Do both the theist and the atheist exhibit faith?

For the purposes of this question I define belief and faith differently:
1) "Belief" is, "A conviction that something is true with or without proof."
2) "Faith" is, "Making a position or taking an action based on a belief."

For example, one may hold a belief that a bridge is strong enough to carry one's weight should one cross over it, and one may demonstrate faith by crossing over it. In these definitions it might be said that, "Faith is belief in action," though that is shorthand and not the main point.

My present understanding is that neither the existence nor the non-existence of God are proven. Yet many people hold tenaciously to the idea that God does exist, and many other people to the idea that God does not exist, and both sets of people make a position of their idea and to some extent or other base their lives on it. Since neither position is proven, under the definitions above may we say that both have a belief (a conviction that their idea is true, in this case without proof) and both have a faith (make a position and take actions based on their belief about this idea). In short, do both the theist and the atheist exhibit faith? And if not, then what is the logical fallacy in the above argument?

This is always a fun one...

For the purposes of this question I define belief and faith differently

This is a problem... if we had the luxury of doing this, we can say that the President is a child rapist, then say for the purpose of this discussion we define "child rapist" as someone who claims to like children. Not to suggest that your definitions are any where near as outrageous, but tf we are making claim that can be damning/offensive to a person or group, we should stick with the common understanding of the terms used. But let's move on...

1) "Belief" is, "A conviction that something is true with or without proof."
2) "Faith" is, "Making a position or taking an action based on a belief."

Then sure, virtually every person on earth exhibits "faith". Example? We all hold a "belief" that 1+1=2, and we all use math regularly in our lives (i.e. acting on "faith" according to this definition).

Yet many people hold tenaciously to the idea that God does exist, and many other people to the idea that God does not exist

(I am going to use Christian interchangeably with "theist" since this seems to be the far more common version of this argument).

The first part is true, as many people profess that God exists (many people will claim to KNOW this, not just believe it). I have yet to meet an atheist who claims that God does not exist. The term atheist refers to a lack of a belief in any gods (an epistemic claim), not a positive claim that no gods exist (an ontological claim). The difference is important.

and both sets of people make a position of their idea and to some extent or other base their lives on it.

Again, I would agree that this is true of the Christian, but not the atheist. Atheists don't base their lives on a lack of a belief in a god any more then the Christian bases their life on the lack of a belief in leprechauns. This idea seems to be based on the common misconception that atheists exhibit immoral behavior because they don't believe in a god. There are many actions the Christian is encouraged to take due to their belief (e.g., prayer, tithing, going to church, etc.) but no such actions the atheist is encouraged to take or takes as the result of their lack of belief.

Since neither position is proven...

Just heading off a potential fallacy here... outside math and logic, things aren't "proven," but evidence is evaluated to help use determine which position is more likely to be true. Assuming just two positions on an issue, the fact neither are proven doesn't mean both are equally likely. This is where evidence comes into play.

So sure, I would agree that according to the definitions you gave, the atheist, theist, and every other person on the planet (with few exceptions) exhibits "faith."

But what if we used more common definitions of faith? This ultimately becomes a messy argument/debate in semantics and definitions. So here is the question I pose to Christians: Is your level of belief in God in proportion to the evidence you have for God? If yes, then there is no room for faith and they are just like the atheist, who lacks a belief in God because the evidence they have for God is also lacking. Of course, many Christians would say that without "faith," they are missing something that God is asking of them (theological claim). If a Christian answers "no" to that question, then says something along the lines of "I don't need evidence, I have faith," then clearly the atheist is not using something the theist is.

Perhaps the atheist believes that life is a result of naturalistic processes devoid of supernatural intervention. To be fair, we should ask the atheist the same essential question: Is your level of belief in this in proportion to the evidence for this? If the answer is yes, then there is no room for faith. Otherwise, perhaps they are using faith.

Short answer: No.

But many won't just accept that, so I'll clarify:

The default position for an atheist toward a claim is not to assume it is true, but to assume it is false until the evidence warrants changing that position. If you were to remove "god" from the logic, and put in "invisible purple unicorn" or "500 pound diamond,"or even "Bujgtniyty," the rational is obvious: the burden is on the claimant to both define what they claim exists (a "Bujgtniyty" has the following characteristics: a, b, c, etc.) and prove that it exists. No rational person would buy a 500 pound diamond sight-unseen, even though it MIGHT exist.

The default position for a theist toward the claim is to accept it not only without evidence, but in spite of contrary evidence. That is literally the meaning of faith: to continue to believe a claim is valid, even if there is a lack of supporting evidence, or if your faith is "trong," despite mountains of evidence to the contrary.

The question attempts to set up a false equivalency between an atheist's demand for proof before accepting a claim, and a theist's demand that their claim be accepted without evidence.

"The default position for an atheist toward a claim is not to assume it is true, but to assume it is false until the evidence warrants changing that position."
This reasoning is known as the fallacy of negative proof........i.e. assume something IS false because it hasn't been proven true