I'm trying to better understand the connection between the concepts of ramification of a field extension, and ramification of a quaternion algebra. I'm also trying to build a better understanding of the motivation behind this term. Bear with me as I summarize my understanding of the stuff...

If $K$ is a field with valuation $v$, and $L$ is a finite extension of it, the definition of ramification depends on whether $v$ is Archimedean. If $v$ is Archimedean, then it corresponds to an embedding $\sigma:K\rightarrow\mathbb{C}$, and also $v$ extends to a valuation $w$ on $L$. We say the extension is ramified here iff there is a nontrivial $\tau\in Gal(L,K)$ such that $w\tau=w$ (turns out to be independent of choice of $w$, hence well-defined). If $v$ is not Archimedean, then $v$ corresponds to a prime ideal $\mathcal{P}$ of $R_K$, and there are prime ideals $\mathcal{P}_i$ of $R_L$ such that $\mathcal{P}R_L=\Pi_{i=1}^r\mathcal{P}_i^{e_i}$, and we have that the extension is ramified here iff $\exists i: e_i>1$.

If $A$ is a quaternion algebra over the field $K$, and $v$ is a valuation on $K$, let $A_v:=A\otimes_KK_v$, and then we have (after a good amount of work) that $A_v\cong M_2(K_v)$ or $A_v$ is the unique division algebra over $K_v$. And the definition of $A$ being ramified at $v$ is that $A_v$ is the division algebra and not the other one.

Now, behind these definitions, there is a lot of other theory one would use to actually determine these things. Let's stick to number fields for simplicity. For a number field as an extension of $\mathbb{Q}$, the infinite places are very simple, just have to do with whether an embedding is totally real. And for the finite places you've got Kummer's theorem: look at when the minimal polynomial of the primitive element has roots modulo $p$, which I find very enlightening because it gives you an explicit description of how $(p)$ splits in the number field. In the case of the quaternion algebras, any $A$ has a Hilbert symbol $(\frac{a,b}{K})$, and you can figure out where it ramifies by looking at $a$ and $b$. The infinite places depend on the sign of the images of $a$ and $b$ under the embedding, and the finite places depend on whether or not $a$ and $b$ (and sometimes $-a^{-1}b$) are in $\mathcal{P}$ and $\mathcal{P}^2$.

So I get all that, but what exactly are we trying to measure here? The infinite places seem more obvious: we are looking at whether or not there are embeddings in the bigger field that look the same in the smaller field. But why should it be so important that the prime factorization of a prime in the bigger ring has repetition? Why is it not called "ramified" just for the prime to split apart into several primes? Also, for $A$ to ramify at a finite place $v$, does this imply some kind of factorization of lifts of prime ideals? $A$ doesn't have a "ring of integers," but it does have an order, does this happen there?

I apologize for the vagueness of my questions, I'm just looking for some good intuition about this from someone who knows it very well. My intention is to understand enough of this for now to use it for something else, but I would like to come back to it periodically until I've mastered it.