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We must draw a distinction between atheism and agnosticism. Atheism is not believing in the existence of a God (or Gods), regardless of whether conclusive evidence is available, while agnosticism is the view that conclusive evidence for whether a God or Gods exists cannot exist. There can be an agnostic atheist, a gnostic atheist, an agnostic theist, or a gnostic theist.

You seem to be claiming that because we cannot have conclusive proof that God does not exist, therefore atheists are operating out of a purely faith-based position. Here's the thing: most atheists are agnostic. They accept that we can never know for sure that God does not exist, but they think it is more likely for the default state (no God) to be true in the light of insufficient evidence by theists. There is nothing wrong with this.

For more about the distinction between atheism and agnosticism, see The Difference Between Atheists and Agnostics by ThoughtCo and the entry on Atheism and Agnsoticism in the Stanford Encyclopedia of Philosophy.

Alvin Plantinga, a Christian philosopher, presents a similar question regarding an extreme form of atheism that he calls "naturalism". Rather than asking whether the atheism of naturalism is faith-based, he asks whether naturalism might be a "religion" (page 311):

Now it is not clear that naturalism, as it stands, is a religion; there is enough vagueness around the edges of the concept of religion for it to be unclear whether naturalism does or doesn't belong there. But naturalism does serve one of the main functions of a religion: it offers a master narrative, it answers deep and important human questions. Immanuel Kant identified three great human questions: Is there such a person as God? Do we human beings have significant freedom? And can we human beings expect life after death? Naturalism gives answers to these questions: there is no God, there is no immortality, and the case for genuine freedom is at best dicey. Naturalism tells us what reality is ultimately like, where we fit into the universe, how we are related to other creatures, and how it happens that we came to be. Naturalism is therefore in competition with the great theistic religions: even if it is not itself a religion, it plays one of the main roles of a religion.

If being faith-based means to believe in a master narrative that answers Kant's questions, that atheism might be considered, using Plantinga's argument, a quasi-faith-based or a quasi-religion. However, not all atheists need be labeled as believing in naturalism or any other master narrative.

There also may be good reasons not to link atheism in general, or even the atheism of naturalism, too closely with religion. In Where the Conflict Really Lies Plantinga wants to show that traditional theistic religions have at most a superficial conflict with science while naturalism has a deep conflict with science through his Evolutionary Argument Against Naturalism. He would likely want naturalism to stand apart from traditional religions when making such a critique of it and not consider it as a religion.

Here is the question:

Given the above, I claim that both atheism and theism are positions based on faith. Would it be incorrect to claim that?

If the atheism being considered presents a master narrative in competition with theistic religions it could be considered a quasi-religion because of that narrative. Not all atheists have such a narrative nor believe in naturalism. Some atheists may be simply indifferent to such narratives.

Plantinga, A. (2011). Where the conflict really lies: Science, religion, and naturalism. OUP USA.

Both positions, the theist and the atheist made a claim: The theist claimed the existence of god, the atheist claimed the non-existence of god. History shows: Neither of them could prove his claim.

Having learned the lesson, today’s atheists make a weaker claim: The traditional god-concept - god being omnipotent, omniscient, all-good - is inconsistent. The concept leads to the problem of theodicy.

Hence atheists consider the existence of god a hypothesis. They claim that this hypothesis creates more problems than solves existing problems. Therefore atheists dismiss this hypothesis. They know: A world-model without a god-concept is less complicated but leaves open fundamental questions due to lack of reliable answers.

IMO that’s not faith but heuristics.

No, atheism is not a faith based position. This has been debunked time and time again and there are numerous resources on the internet that cover this error in logic. If theism claims the existance of a god, an atheist is one who is not convinced of that claim and rejects it.

An atheist has no responsibility to disprove the claim that god does exist because the atheist is not asserting anything. The burden of proof is on the person trying convince another with their claim. Claims require support and the thiest made the claim so the thiest must support it. Not having faith in something is not faith much like not having a hobby is not a hobby and not exercising is not a form of exercise.

The crux of the matter is neatly hidden within the definition of the word faith. Mind you, both Cambridge Dictionary and Oxford Dictionary say that the main meaning is:

Complete trust or confidence in someone or something.

‘this restores one's faith in politicians’ [...]

and

great trust or confidence in something or someone:

She has no faith in modern medicine. [...]

As you can see faith is a thing that can change on the spot. But many people define faith as something fundamental and unchangeable, that cannot be impacted by such trivial action as observing the world outside. Thus a 'loaded term' complicates the discussion. Let's resolve it by naming the latter thing an axiom (unchangeable) and the former a belief (very much changeable).

If you assume axioms, you can base proofs of them. These proofs expand your knowledge about the world created by the axioms. The proofs don't depend on observations of the world, but on observations of the axioms. This is your current worldview, as your question mentioned words 'proof' and 'proving' about six times.

On the other hand, beliefs don't enable proofs. This is scary. One day you believe electrons are small chunks of matter, tomorrow some irritating publication appears about how an electron was observed to interfere with itself and you have no authority to banish these outrageous observations. Then they build a bunch of stupid transistors and they somehow work and beautify our lives, although nobody proven that beforehand.

Atheism is overwhelmingly a belief. (Or it can be based on axioms, if something goes terribly wrong.)

Theism is overwhelmingly based on axioms. No major religion says "If this book doesn't work too well in practice and you see other theories working better - then by all means use them! Absolved! Amen!".

Answers so far have considered evidence, proof and strength of argument. There's another point to consider though - utility. What is the purpose of these claims?

If the theist considered their god(s) to be a non-participant in the world and any hypothetical afterlife, then it would not matter whether the god(s) existed or not. The debate simply wouldn't exist. A debate only exists because theists claim knowledge about their god(s)' interaction with the world and/or actions in the afterlife; and thereafter knowledge of what must be done to ensure people gain the favour of the god(s) for this.

This leads us to the Atheist's Wager, which proves that regardless of the existence or non-existence of any god(s), the best outcome in both the temporal and spiritual domains is to live your life as if no god exists, and live a good life as defined by humanist philosophy. Not only that, but it also provides the best outcome for other people, and hence is the best moral position. This is proven by simple logic, without the need for evidence of the truth of either case.

Having proven that humanist philosophy is the highest moral standard and the best outcome in all cases, it's then necessary to question why we should care whether the god (s) exist or not. This is the crux of where the theist's case falls down. When the existence or non-existence of god (s) is seen as a key question for how to live your life, of course it's important. But take that away, and the theist's whole argument is no more relevant than a 5-year-old trying to get their parent to answer whether they think a ninja could beat a dinosaur.

Of course having an opinion on the subject is based on faith. But being willing to argue on the subject is based on either being irrational or having too much free time. As such, the fact the theist even cares about it invalidates their argument.

You already noticed, that theres different "levels" of atheism. (agnostics)
I want to show yet another perspective, that doesn't rely on faith.

Alot of Religions, are not worth following, even if you assume everything they say is correct.

If you get to heaven even without following the religion / the god is allforgiving, then there is no need to follow it.

If you don't get to heaven despite living a moral life but not believing in god, then God doesn't seem like he is worth following. What god would condem someone to hell, because he uses his mind that supposedly god gave to him.

If God is like he is described in a lot of the big religions, then he doesn't seem worth following, just think about what the Christian god has done in the bible. (Mass extinction of innocent people with a flood, telling someone to sacrifice their son, and so on).

You might even argue, that if God is omnipotent, then he is not very nice for letting innocent kids die horribly.

All of these positions have nothing to do with faith. You don't believe (or disbelieve!) in anything, you just look at the possibilitys and act based on logic and reason.

You might also reject the Idea of following a Religion, because from a pure numbers game, you are very unlikely to be right. If we assume that a religion is based on faith, then you can't choose one religion over the other. So what now? If you are in a Religion you are still an "atheist" to all the other religions. How can you reconsile with that?

It's very simple. Theism is belief in deity. Atheism is lack of belief in deity.

Neither position comments upon the ontological status of deity.

Atheists and theists, however, may hold any variety of opinions or beliefs.

The only requirement of meeting the definition of "atheist" is that you do not have belief in any god(s). Computers, rocks, and newborn babies are all, by definition, atheists; because they do not believe in any god(s), notwithstanding that they do not understand what is meant by "god", nor that they have never even thought about the question, nor that they do not even have the cognitive capacity to consider the question.

No faith is required to not subscribe to a faith-based belief, because having faith is not a default attribute of anything, human or otherwise. It is simply not part of the definition of being an atheist that you must have some kind of belief one way or the other on whether there is one or more god(s).