The two definitions you’ve given both apply, and in fact are really the same word with two slightly different senses. They would both be listed as variants of the same basic word.
Understand that being so short, nip is a very old and very common word that has historically been put to a multitude of unrelated uses in English. The OED lists at least nine different words nip, plus offers it as an archaic spelling for neap, too, plus comes in the gnip and knip variants itself. (Another sense of nip is in catnip.)
For the particular verb nip in question here, the OED offers these distinct senses, amongst many others:
- trans. To compress or catch between two surfaces or points; to pinch, squeeze sharply, bite.
- To sever, remove, or take off, by pinching.
- To defeat narrowly (in a sporting contest). U.S.
- To check the growth or development of (something), after the manner of pinching off the buds or shoots of a plant. Also with off.
- Of cold: To affect (persons, etc.) painfully or injuriously.
- To touch or concern (one) closely; to affect painfully, to vex.
- To snatch, catch, seize or take smartly. Also with away, out, up.
- slang. To arrest.
- intr. To give a nip or pinch; to cause or produce pinching.
- slang. To move rapidly or nimbly.
A “love nip” is a light touch. It may be related to the use of the noun nip in the game of Cricket:
- A slight touch or stroke given to the ball by the batsman; a tip.
Or to this other noun use:
- A small portion, such as may be pinched off something; a fragment, little bit.