Deciding on mathematical notation

Some thoughts, too long for a comment.

The answer to your question will depend in part on the audience. Are these notes for you for future reference, or do you hope or plan to share or publish them?

I'm inclined to suggest that you be conservative. When established notation exists you should have a good reason for replacing it.

New temporary notation to make nearby formulas shorter and easier to parse is fine. New notation defined at the start of a long document may make your readers jump back and forth between definition and use.

If your notes are (or will be) written in $\LaTeX$ then you can write macros for the things you might want new notation for. That way you can use your private language in the source files and postpone deciding how it should look (standard or newly invented) in the final text.

Clear and concise mathematical notation is often hard to come up with, and often follows the introduction of a theory by many years. The original person to come up with an idea may not also come up with the best way to write it out.

A great example is Maxwell's equations. If you read his original papers, he uses component notation and although the equations exhibit a tantalizing pattern, they are nowhere near as clear as the now-familiar equations in modern notation.

The skill of developing good notation is related to, but not completely correlated with, the degree of math insight or ability to solve difficult problems.

For your question about integrals, usually it is best to leave things with the integral signs. The major exception is when doing a calculation that finds the value of an integral by squaring it or showing that it is equal to some constant minus itself. For example, a familiar proof starts with:

Let $$ I = \int_{-\infty}^\infty e^{-x^2} dx $$ Then $$ I^2 = \int_{-\infty}^\infty e^{-x^2} \int_{-\infty}^\infty e^{-y^2} dx \,dy =\int_{r=0}^\infty\int_{\theta=0}^{2\pi}e^{-r^2}r\,d\theta\,dr = \left. 2\pi \frac{e^{-r^2}}{2}\right|_0^\infty=\pi $$