These are all questions I've had myself, so hopefully I can help and not confuse you more:
So, all other things being equal, when A is in B's domicile or exaltation (and they are in aspect) it is not A that loves B because A is in B's domicile/exaltation (as Frawley would have said in his books and magazines) but B that receives A (according to the traditional astrologers) - hence it is B that likes/loves A. Am I correct?
All things being equal, that's right. You would say A planet receives B planet when B is in a sign of A's domicile, exal, trip, or two lesser dignities. The way I think about it is that planet A wants to be where planet B is because that's an area of the zodiac where it has essential strength. If you study hyleg/alcoccodens it starts to make more sense too because an alcoccoden has to be the planet with the most strength in the place of a hyleg and aspect it (the alcoccoden receives the hyleg).
Robert Hand has a useful workshop on YouTube that you can watch called, "Reception." Normally you'd want reception to be both ways, in traditional, for it to lead to much, but one-way can be descriptive.
AND, again, remember the condition of the planet that is trying to recieve another. Say you have a chart where Venus and Mars are the querent/quesited and Venus is in Scorpio and Mars is in Taurus in a square aspect. Squares are of course difficult already, but each planet is in the sign of their own detriment while the each are in eachothers' domicile. So does Mars receive and love Venus and Venus receive and love Mars because they are mutually in each other's domicile? They try to receive/love but can't properly because they are in detriment themselves and a square is a conflictual relationship.
One other thing. When A is in B's domicile, B is also dispositor of A which means that B has "power" over A. In this case, does B show more power (because of dispositorship) or love (because of reception). I understand that reception requires aspect - does the having power over a planet by dispositorship also requires aspect (which then mean that reception is exactly identical to reception)?
There is never an easy answer to such questions because it depends on factors that lead to a whole. I think I've heard the dispositor described as a "host" the most, though some have called it a "landlord". The dispositor (host) has a vested interest in making its guest happy and try not to hurt it. Reception is part of this obviously because the host receives its guest, so it cares for it, watches out for its interests, or loves it, etc. If there is a proper aspect, that is more significant because the host can "see" the guest. If the host is a crazy malefic, it can be a good thing when it is unable to cause harm to you, but nobody wants a crazy host because you are somehow connected to them - the landlord concept makes this easier to envision. Landlords usually do hold power, right? In "The Reception Debate" link I posted, Deborah talks about a Sun in Libra - Mars in Leo aspect, a sextile, and she mentions that Mars feels overwhelmed by the Sun because he is Mars' dispositor and does have control over him, but this is also because the Sun brings a sextile aspect to Mars, which is also the place of Mars' detriment in Libra. So, yes, the dispositor (or host) can hold power, but if there is an applying aspect by the host this is more so. As in her example, the dispositor can sometimes cause harm to the planet that is its guest, even when it doesn't intentionally want to. The "love" nature would have to be analysed by the reception, aspect and dignity of the planets in concert because each one would give a clue about the whole. Since Sun in Libra is in fall, he is unable to receive Mars even though Mars is in Sun's domicile of Leo, and Sun is also afflictive because, again, Sun is bad off and also brings Mars to a place where he is weakest in Libra. They both likely care about one another because of the sextile alone and also because Mars is in Sun's domicile, but caring for each other and being a good influence are two different things. You normally don't hurt people when you truly love them, although it can be unintentional.
In the "Reception" conference, Hand talks about a "pushing dignity" concept too, which is when an essentially debilitated planet can receive strength from its dispositor if the dispositor has essential strength itself and aspects it. So the host is a source of strength for its guest.