Your summary seems not quite accurate.
In reality, there is variation, not a single absolute rule. Supposedly, some Spanish speakers even distinguish a [b] sound from a labiodental [v] sound—for example, if they speak another language where such a distinction exists.
But for most Spanish speakers, there is no phonemic distinction—meaning no possibility of meaningful contrast—between [b] and [β]. This means it's not possible for "words loaned from other languages, foreign names, abbreviations and similar stuff" to follow a different rule from regular words.
The general pattern, which I've been told is advisable for learners, is is simply to use [b] after a pause or after a nasal consonant (the preceding nasal consonant will be pronounced [m], but can be spelled M or N, and as you note, doesn't have to be in the same word as the [b]), and to use [β] anywhere else. So:
- «b» or «v» is the first letter of the sentence's first word;
Yes if this is the first sentence that is being said after a pause. If multiple sentences are being pronounced one after another, the B or V at the start of one of the middle sentences might be pronounced as [β] if the speaker doesn't pause before it.
- «b» or «v» is the first letter of a word and there is comma before it (so, pause is unavoidable);
I don't think people really always pause after a comma. If you do pause, then you can say [b]; but if you don't pause, it could be [β] (or [b] in the cluster [mb]). It would depend not on how the sentence is punctuated, but how it is actually pronounced.
- there is either «m» or «n» before «b» or «v» (it includes the case when the word starts with «b» or «v» and the previous word ends with «m» or «n», like «un vaso»);
Yes, that's correct.
- words loaned from other languages, foreign names, abbreviations and similar stuff.
No: as I said above, these would follow the general rule for [b] vs. [β] (assuming the speaker is pronouncing the loaned words/named in Spanish and not "code-switching" to another language entirely: in the latter case, anything goes, and not only [b] and [β] but other sounds foreign to Spanish could well be involved).