Recognizing though that method of understanding something is pretty personal, I will say that what really helped me refresh was the 3Blue1Brown Essence of Linear Algebra series on Youtube. If you're trying to better grasp the subject, do yourself a favor and check out those videos.
I especially enjoyed this video (not exactly a math tutorial more of a cool explanation): https://www.youtube.com/watch?v=OkmNXy7er84
In particular his recent "Lockdown Math" series taught me a lot.
I knew and used trigonometric functions and exponentials before but they never really "clicked".
Grant is a tremendous educator.
https://youtu.be/ZxYOEwM6Wbk?t=481
Is this really the case? I checked Wikipedia and MathWorld and nobody makes a distinction between e^x and exp(x).
Even if it's a "white lie" for didactic reasons, I don't buy it, it will be much more confusing down the line for students.
Math is about finding structures governed by some rules and then generalizing them. To me the e^x defined as repeated multiplication conceptually is the same thing as the exponential function, just over broader domain. Why? You can interpolate between integer-valued X's using geometric mean. e^3 = sqrt(e^2 * e^4). What stops you from interpolating this recursively to achieve in the limit the exponential function over real numbers?
Of course, you can also go another route: you can define the number e (e.g. as the limit of (1+1/n)^n), then you can straightforwardly define natural number powers of e by repeated multiplication, then integer and rational powers through reciprocals and roots (you have to prove n-th roots exist, but that's doable), and then you can define real number exponents via limits (maybe this is what you mean by "interpolating recursively"). Now, when we come to complex numbers, you can use Euler's formula as a definition instead of a theorem, and define exp(a+bi):=e^a * (cos b + i sin b). Of course, at the end of this whole exercise you can prove that this definition is exactly equal to the power series definition.
Another benefit of the power series definition is that it also generalises e.g. to the matrix exponential (exp(A), when A is a matrix).