Better understanding dividing and forking Recently I learned about dividing and forking of formula / partial types:
We say that $φ(x,b)$ divides over $C$ (where $b$ in the monster and $C$ is small) if there exists an indiscernible sequence $\{b_i\}_{i\in\omega}$ over $C$ of elements with the type $\operatorname{tp}(b/C)$ such that $\{φ(x,b_i)\}$ is inconsistent.
We say that $φ(x,b)$ fork over $C$ if it implies finite disjunction of dividing formulas over $C$.

I am trying to get a better intuition about those definitions, especially about forking.
From my understanding, $φ(x,b)$ divides over $C$ if $φ(C,b)$ (i.e. the set $φ$ defines with $b$) is not "big" in the sense that for every $A\subseteq ω$, $|A|=k$ (for some fixed $k$) we have $\bigcap\limits_{i∈A} φ(C,b_i)=∅$
This view is reinforced when we see that the forking formulas are basically the ideal generated from the dividing formulas (I mainly worked in set theory with filters which are "defining big sets", so my intuition about their dual, the ideals, is "defining the small sets").
But this view doesn't feel that good to me, it is somewhat lacking: specifically, what is the role of the second parameter ($b,b_i$) here, why do we look at what happens to the defined subset when we change it. And why are we fixing $C$? Isn't the first parameter is usually the parameter we move around?
Further more, I feel like this view falls completely for forks: the set of forking formulas is not always a proper ideal, even over the empty set (e.g. in $(X,\mathcal P(X),∈)$ or circular order, in which $x=x$ forks over $∅$)
My understanding on forking/dividing partial types is basically the same as my understanding of forking.


So I am guessing that my question is: is there a standard way to think about dividing and forking formulas and partial types?
Also, I focused a lot on the study of set theory/formal logic, and kind of neglected abstract algebra for some time (which comes to bites me in the butt now...), so if there is a view that appeals to set theorists and relies less on algebraic intuition I would love to hear about it.
 A: Non-forking and invariance (a notion easier to grasp) coincide in a large class of theories.
Below I elaborate with a model $M$ instead of a set $C$ to avoid some technical issues (b.t.w., these technical issues are quite interesting, just not now).
Things are clearer when working with global types.
We say that $\mathscr{D}\subseteq\mathscr{U}$ is invariant over $M$ if $\mathscr{D}$ is (setwise) fixed under the action of Aut$(\mathscr{U}/M)$.
To a global type $p(x)$ and a formula $\varphi(x,y)$ we associate the set $\mathscr{D}=\{a\in\mathscr{U}:\varphi(x,a)\in p\}$.
The global type $p(x)$ is invariant if all sets $\mathscr{D}$ as above are invariant (as $\varphi(x,y)$ ranges over all formulas).

Theorem Assume $T$ is NIP (the definition of NIP is irrelevant - just a large class of theories that include all stable theories) then the following are equivalent for every global type $p(x)$

*

*$p(x)$ does not fork over $M$;


*$p(x)$ is invariant over $M$.

See Theorem 5.21 in P.Simon's book
For an easier comparison with non-dividing, note that a simple argument of compactness proves the following

Fact For every $T$ the following are equivalent for every global type $p(x)$


*$p(x)$  is invariant over $M$;


*$\varphi(x,b)\in p$ and every $b_1\equiv_M\dots\equiv_Mb_n\equiv_Mb$ there is a $c$ such that $\varphi(c,b_i)$ for every $i$.

It is evident that non-dividing is a weak analogue of 2.
What about non-NIP theories? What about non-global types? Can we say something meaningful about 2 and/or 4?
I wish I could answer YES. But there is a practical problem. The notions in 2 an 4 are semantic in nature. Though we (human being) can figure out semantic better, we mainly can work with syntactic notions. Dividing and forking offer a syntactic hand-grip.
Between a natural notion with few theorems and a unnatural one with many theorems mathematicians clearly choose the second (who would disagree).
A: Some years ago I wrote an answer about the motivation for the definitions of dividing and forking. You may find it helpful: https://math.stackexchange.com/a/1613033/7062
But I'll add to what I wrote there, addressing some specific points in your question.
You wrote:

From my understanding, $\varphi(x,b)$ divides over $C$ if $\varphi(C,b)$ (i.e. the set $\varphi$ defines with $b$) is not "big".

That's exactly the right idea, except that we shouldn't think about the set $\varphi(C,b)$, but rather the set $\varphi(\mathscr{U},b)$, where $\mathscr{U}\models T$ is the monster model. Often $\varphi(C,b)$ might be empty (as small as possible!) even though $\varphi(x,b)$ does not divide.
In fact, to start getting intuition for these notions, it's probably best to think about the case $C = \varnothing$.
If I give you a definable set in $\mathscr{U}$, say defined by the formula $\varphi(x,b)$, I can also find "copies" of this definable set that "look the same". What makes a copy "look the same"? Well, it should be defined by the same formula, using parameters that "look the same": $\varphi(x,b')$, where $\mathrm{tp}(b) = \mathrm{tp}(b')$. By the strong homogeneity of the monster model, $\mathrm{tp}(b) = \mathrm{tp}(b')$ if and only if there is an automorphism $\sigma\in \mathrm{Aut}(\mathscr{U})$ such that $\sigma(b) = b'$, and in that case $\sigma(\varphi(\mathscr{U},b)) = \varphi(\mathscr{U},b')$.
Intuitively, we say a formula divides (over $\varnothing$), if it has a lot of copies with very little overlap. In the answer linked above, I tried to give some intuition for why this is a good notion of "smallness".
Now it's often useful to draw finer distinctions for when two tuples "look the same" than looking at their type over $\varnothing$. We could instead fix a set $C$ in the background and insist that a "copy" of $\varphi(\mathcal{U},b)$ should be defined by $\varphi(x,b')$, where $\mathrm{tp}(b/C) = \mathrm{tp}(b'/C)$. Equivalently, these definable sets are conjugate by an automorphism of $\mathscr{U}$ which fixes $C$ pointwise. This is the role of "over $C$" in the definition of "dividing over $C$".
Whenever you think about dividing or forking over $C$, you could just as well think about dividing or forking over $\varnothing$, in the expanded language where we name every element of $C$ by a constant.

This view is reinforced when we see that the forking formulas are basically the ideal generated from the dividing formulas (I mainly worked in set theory with filters which are "defining big sets", so my intuition about their dual, the ideals, is "defining the small sets").

That's exactly right: once we have a basic notion of "smallness" (dividing), it's natural to want the small sets to form an ideal in the Boolean algebra of definable sets. The forking formulas are exactly the ones which are "small" in the sense that they are in the ideal generated by the dividing formulas.
From a technical viewpoint, the fact that the forking formulas form an ideal lets us take any non-forking partial type (that is, any filter in the Boolean algebra of definable sets which does not meet the forking ideal) and extend it to a non-forking complete type (an ultrafilter which does not meet the forking ideal). In practice, when $C\subseteq B\subseteq A$, this is used to take a type $p(x)\in S_x(B)$ which does not fork over $C$ and extend it "generically" to a type in $S_x(A)$ which does not fork over $C$. In other words if $p(x)$ doesn't fork over $C$, we can extend it to any larger set of parameters without being "forced" into any definable sets which are "small" from the point of view of $C$.

I feel like this view falls completely for forks: the set of forking formulas is not always a proper ideal.

Well, it's a common theme in mathematics that attempts to distinguish between large and small sets don't work in all situations. For example, from the  point of view of Baire category, we introduce a basic notion of "smallness", namely nowhere denseness, and then we want the "small" sets to form a $\sigma$-ideal, so we define meager sets to be countable unions of nowhere dense sets. This works great for some spaces (those with the property of Baire), but in other spaces we find that the entire space is meager, so this particular large/small distinction is useless.
It's quite an amazing phenomenon in model theory that the analogous situation is so rare: there are only a few natural examples where the formula $x = x$ forks, so that the forking ideal is the entire Boolean algebra of definable sets. If we restrict attention to simple theories, we can prove that this never happens, which is one reason why the theory of forking and dividing is so well-suited to the context of simple theories.
