The Terminology of Faith vs. Assumption
First, I will try to explain why the word assumption is better than the word faith in this context. In the definition of faith you have, it ends with "for which there is no proof" and this, ultimately, is the issue. Faith, being defined in that way, means that it applies to things that cannot, under any circumstances, have a proof. However, the axioms we use in mathematics do not fall under that category. Axioms are relatively unprovable. This means that a statement, such as "parallel lines never intersect on a flat surface," might be assumed without proof in one context, but is completely provable in another context. I'll explain this with an example below, but the point to take away is that there are proofs of mathematical statements that we assume as axioms, but those proofs are in systems that do not use those statements as axioms.
The reason why the statement "we take it on faith that our axioms are true" is less true than "we assume our axioms are true" is because faith says "we are saying there is no proof that parallel lines on a flat plane never intersect but we believe it anyway" when assumption says "we are assuming that parallel lines on a flat plane never intersect without proving it". We don't take it on faith that Euclid's postulates are true, we assume them without proof because they are arguably self evident, or even definitional. The difference comes down to the difference between "there is no proof" and "without proof".
An Example of the Relativity of Proof
As I said above, a certain mathematical statement may be assumed as an axiom in one context (in one specific theory) or it may be derivable as a theorem in another context. Let's take for example Euclid's first postulate:
Any straight line segment can be drawn to adjoin any two points (on a flat surface).
If you are trying to develop Euclidian geometry, this is one of the assumptions you need to make. This, along with his other postulates, are assumed without prior proof and are used as the starting points for logical deductions to be made. We start off assuming that a collection of things are true about lines and points and we then use deductive proofs to prove theorems. Some examples of theorems that Euclidean geometry proves (things like the Pythagorean theorem) are found here.
In current foundations of mathematics we use the axiomatic system ZFC set theory. ZFC is also a list of axioms equipped with a proof system that lets us derive theorems, however ZFC is much more powerful than Euclid's postulates. ZFC tells us that sets, a certain amount and kind of them, exist and that those sets have certain properties. The method of using set theory as a foundation for mathematics is done by treating the sets that ZFC talks about as numbers and that the properties that the sets have reflect the properties of actual numbers. Doing this, ZFC is able to prove (almost) all mathematical statements about geometry, algebra, number theory, analysis, and so on. What ZFC cannot prove is its own axioms. However, ZFC can easily prove Euclid's first postulate once a suitable definition of "line" "point" and "straight" are defined in the theory. Doing an actual deduction in ZFC is very tedious; you can see here for the details of as well as the actual proof of 2+2=4 in ZFC.
So hopefully it is clear now why the relative assumption of an axiom in one case doesn't mean it is "without a proof" in all cases. But, this also sounds like it might end up in circular reasoning, so let me show how that is dispelled.
Obviously, if we say "well in one theory we have axioms A and B and those axioms prove C and D, and in another theory we have axioms C and D and those axioms prove A and B, so we have a perfect set of theories!" then we have circular reasoning. While those statements may be true, it would be dishonest to use them to justify each other. We need to pick one theory as a bedrock and have those axioms assumed in all cases, otherwise everything would be circular.
Philosophical Justifications for Axioms
If we agree that it is essential to have a bedrock of assumed axioms that will never be justified in our theory, how do we choose them? In mathematics, especially after the formalization that happened after the foundational crisis around the turn of the 20th century, philosophers, logicians, and mathematicians have spend considerable time discussing the axioms we eventually settled on (ZFC) and how we can justify what they say. Honestly, a lot of these ideas are not new and go back to the ancient Greeks.
A main idea is that the axioms are self evident, meaning that when we think about what they mean, they seem plainly obvious, even definitional. This is what Euclid thought of his postulates. There are translations of the Greek word "axioma (ἀξίωμα)" that include "self evident" in the definition.
Now, take for example the ZFC axiom of extensionality. This axiom states that if two sets have exactly the same elements then those two sets are equal to each other. This axiom, although it cannot be formally proven within ZFC, seems almost definitional to the idea of equality, doesn't it? It also is a perfect candidate for something that is self evidently true. Is it possible for two things to be made of exactly the same things and yet be different? It seems, at least to our rational intuition, that that cannot be true. A contemporary defense of the ZFC axioms, as pointed out in the comments, is Believing the Axioms by Penelope Maddy. In it, Maddy even argues that the axiom of extensionality seems to be the most definitional of all of the ZFC axioms.
The Validity of a Proof
In regards to your final question, the answer is no, there is no relationship between proof in mathematics and faith. Mathematics uses deductive reasoning, in the form of a formal proof system which are objects studied in proof theory. During a mathematical proof, you start with some statement and then apply a rule of inference to gain a new statement. In mathematics, we start with axioms and use our rules of inference to derive theorems, and often times we then can use those theorems to derive more theorems and so on. However, this method of inferring conclusions from premises has nothing to do with faith or with assumptions.
In proof theory, a set of rules of inference can have two important properties: validity and soundness. A valid argument, where argument means a specific instance of some premises and a conclusion, is one where "if the premises are true, then the conclusion must also be true." An example of a valid argument is:
All dogs are purple
Molly is a dog
Therefore Molly is purple
Now, if this argument is about the real world then we know that it is not true. However, it is a valid argument because the conclusion is assured to be true if the premises are true. Soundness is similar, except that sound arguments actually are true. For example:
All dogs are mammals
Molly is a dog
Therefore Molly is a mammal
Soundness is stronger than validity (notice that the sound example is also valid). The proof systems that are used in modern mathematics are not only valid, they are sound. It is precisely why we choose to use them, because it is a perfectly mechanical process to test whether or not a proof system has those properties. It has nothing to do with faith or assumption, those properties are things that can be shown. The only assumption we need to make is what axioms we pick, and there is a long history of philosophical arguments as to why the axioms we have picked are good axioms to believe in. The proof is the part that does not require assumptions; choosing axioms is the only step that does.