Lastly, I want to it still leaves open the last bulleted question, which is an objective one, is still open.

The former leads to all kinds of trouble if one introduces various derived concepts - but for an ad-hoc definition works fine. I also have added another book besides Sonderman for comparision, if you would share your opinion on that, I'd been interested :).

@KurtG. Thank you for your input, I will consider that. I actually don't mind long-winded intros, because I want to really know the details well and I feel many books skim over them too quickly to get to the interesting parts, disregarding that some "intellectual debt" invariably builds up which might turn out later to be problematic (though I'm not saying that Sonderman might suffer from this, since I haven't read him). A good analogy is introducing curves as maps, instead of equivalence classes of maps that have the same parametrization. [...]

This makes the difference perfectly clear, thank you. Though now I wonder why this difference runs under the description " dynamical perspective", since it seems to me to be a purely algebraic issue...(do you have any ideas concerning this ?)

I was hesitant to ask this here, instead of math.SE, since the question seems to elementary, but have finally decided to ask it here, since (1) it orginated from a question from this site and (2) if the author of the statement doesn't know the difference (although he may have been jocular when he said it) an answer to this question may also be worthwhile to other algebraists on this site.