In many ways, I am atypical in the way that I approach a problem, but it works for me. Specifically, I try to understand an example in as much detail as I possibly can. If the example, is too complicated, then I make a simpler example. As much of the intricate detail that I can bring to bear on the example is brought.
For example, instead of trying to understand Lie groups and Lie algebras in general, start with the circle and the line that is tangent at the point (1,0). What is the exponential map? Oh, OK. Now how about $SU(2)$ and $su(2)$? Can you understand that the Lie group is the $3$-dimensional sphere? Can you understand the coordinates? Can you understand the equators? How do $i,j$ and $k$ really work?
What is the difference between the multiplication rule $i\times i =0$ and $i^2=-1$?
I spend time pondering. And often my notebooks will contain tangential problems or specific computations. I will keep doing the computation until I get it right! If necessary, I will write a program to complete the computation. When I understand the example completely, it is usually easy to abstract.
Then I follow up, usually writing in a notebook or several notebooks before I begin writing on the computer. I have an advantage in that I have long-distance collaborators, so it becomes necessary to explain the idea to the collaborator(s). That is the first writing stage: write for someone who knows your short-hand and your metaphors. the second stage is to write for someone who does not. Then I write with a set of colleagues in mind, but I assume the colleagues do not remember anything from the previous work. I also try to explicate the notation writing for example "the function $f$, the knot $k$, or the tubular neighborhood $N$.
A complex analytical colleague only uses $z$ for a complex number, $x$ for a real variable, and $n$ for an integer. These variable choices are culturally determined, and so one keeps with the culture of the discipline unless there is good reason to deviate. As a final example of this, the variable $A$ in the bracket polynomial is known to everyone in the field. The variables $q$, $t$, $X$ etc. are less known and involve different normalizations. So it is the burden of the author to relate these to the more well known choices.
Perhaps the best way to think about the difference is as one of emphasis.
"Proof-based Calculus" would have the goal of doing Calculus but would justify the methods of Calculus (integration and differentiation) by proving their validity.
"Analysis" would have the goal of developing the theory of Calculus (and more than just Calculus)* at least partly for its own sake.
Consider the example of the real numbers. "Proof-based Calculus" is concerned with the real numbers only because it will need them for doing Calculus. "Analysis", on the other hand, treats the set of real numbers as an object worthy of study without having to consider further applications.
*Analysis is also much broader in scope than Calculus.
Best Answer
I write $\in$ (
\in) with two strockes, as a lowercase 'c' followed by a horizontal cross-bar. I write $\varepsilon$ (\varepsilon) in one stroke like a backwards $3$. I tend to avoid the symbol $\epsilon$ (\epsilon) entirely. Perhaps the following images will be helpful: