A series of unfortunate events made the very useful technique of reception very confusing. First I really regret that the 17th century astrologers messed with it and tried to limit it to mutual reception, then some contemporary traditional astrologers further muddied the waters.
Let's start with the traditional definition from Bonatti, the classic medieval source,
"And it is said if a planet is joined with the lord of the sign in which it is or the lord of the exaltation of the same sign, or with the the lord the [term] or triplicity or face, either bodily or by aspect [they are in reception] However, this is true [only] if it is joined with the lord of the [sign] or exaltation, or with the lord of two of the lesser dignities, to wit with [a planet which is] the lord of the [term] and triplicity, or with lord of the [term] and face, or with the lord of the face and triplicity..."
Liber Astronomiae, Tr. III ch 13 (the corresponding passage can be found in the Dykes trans at 223)
In other words, you need two planets and (1) they are making an applying aspect and (2) at least one planet is in the other's sign or exaltation, or in the others triplicity plus term, triplicity plus face or face plus term.
Even though Lilly only defines reception in terms of mutual reception, ie two planets, both in each other's sign or exaltation, he actually uses the older definition in examples in Christian Astrology. I much prefer the term "generosity" which Ibn Ezra uses for two planets each being in each other's sign or exaltation without needing an aspect. Beginning of Wisdom, Epstein trans page 125.
Then to make things even more confusing some contemporary traditional astrologers starting saying that planets in mutual reception switched signs or places or something, what? I never understood what this was about and it is not supported by actual traditional sources.
So with regard to reception we have a big mess! Actually it is pretty straightforward. If you have two planets making an applying aspect and at least one is in the sign or exaltation or two minor dignities of the other, it makes the aspect go much more easily. A square with reception means problems that can be worked through, for example, and a trine with reception means smooth as silk! See Bonatti, Liber Astronomiae, Tr 6 Horary ch 2 Dykes trans page 362-4 for examples of this. If you don't have a copy of Bonatti's Horary Section of LA GET ONE! Here it is as an Amazon paperback Dykes has even put this out as an e-book.
They don't switch places, they don't increase in essential dignity, the interaction between them just goes so much more easily. For example, if Mercury is the ruler of the 7th applying to Jupiter, ruler of the 1st and Mercury is in Cancer and Jupiter in Aries, they are making a square with reception since Mercury is in the exaltation of Jupiter. However, Mercury is still peregrine and Jupiter, if this is a night chart, just dignified by triplicity, as the night ruler of fire signs. They aren't increased in essential dignity by reception, thus they aren't any stronger, but that argument they are about to have, maybe they can work through it.
Reception is a good example of where I part ways, even with Lilly, when that "modernizing" impulse kicks in. This was a problem even in the 17th century!