I read your response like five minutes after you posted it as I just replied to cbzbc at that time. Thanks a lot for the effort! It's already pretty late here, so a real response will have to wait till tomorrow, but I guess I was able to get quite a few things from your responses. Especially probably somewhat related to your second point in the last comment, namely that I kind of really want to avoid the math and use analogies but without realizing or knowing well enough where those analogies break. Not because I don't like math but because as a layman I mostly consume it passively when reading or watching lectures. In consequence I can follow a presentation but doing calculations myself would be a very different thing and so I also lack mathematical intuition when it comes to the math used in physics.

And as I am already typing, I will just add the first question I got when reading your response because I already typed it out. In the second paragraph - not counting the first sentence - you write »[...] \eta is the chosen background field, h is the perturbation field, and g is the "true" metric.« When you say »chosen«, are you only referring to the choice between signatures (1,3) and (3,1) or do you want to allow eta to be something other than the Minkowski metric there? Probably not to important overall but the first thing that I marked.