Partial answer to share what I've found to date:
The term "merger" does have widespread use in galactic-dynamics. The word "merger" appears 71 times Wikipedia's Galaxy merger for example, with terms like binary merger, multiple merger, minor merger, major merger, wet merger, dry merger, damp merger, mixed merger, merger history trees all having explicit definitions there.
Supermassive Black holes
When galaxies merge, there is the question of what happens to the supermassive black hole (SBH) that may be in the center of each. Since galactic mergers and SBH mergers are inextricably linked I'll list their questions together here:
But for individual stellar objects the situation does get murky. We have had a collision tag for a while now, and the "merging" of stars due to collisions happening in the centers of dense clusters is a topic first raised decades ago. Whether they merge, or just exchange matter or something else, it seems infinitely safer to stick with collision.
But as @DaddyKropotkin points out:
We're just giving you basic sense of how the terminology is used. It is subject-specific and nuanced since the word "merger" is used in the new field of gravitational wave astronomy, so you could get many different, inconsistent, opinionated answers.
When two objects that are either black holes (BH) or neutron stars (NS) find themselves in extremely close proximity, usually through a process of orbiting each other and spiraling inward due to energy radiation in the form of gravitational waves, and ultimately touch and combine much/most of their masses to form a single object, the last few seconds generates gravitational waves so strong that we can detect, record, and analyze them. These are then also called "mergers" (BH-BH, BH-NS and NS-NS mergers) See Wikipedia's List or gravitational wave observations; List of gravitational wave events for example.
From Appendix A: Astronomical Terminology:
Energies of the order inferred suggest that gamma ray bursts may originate in the merger of two neutron stars to form a black hole or the capture of a neutron star by a black hole. Such mergers provide almost the only ways in which we can conceive of vast amounts of energy to be liberated rapidly. The potential energy that can be released in these mergers is of order M0c2 ~ 1056 erg.
As for stars merging:
What can be said?
- "Merger" is a solid, standard term when it comes to galaxies and their supermassive black holes.
- "Merger" is becoming a standard term for the last few moments of NS-NS, NS-BH and BH-BH grabitational wave event.
- In the case of stars, it's murky and "collision" seems to at least adequately cover all possible types of events where there is substantial combination of two stars' mass into one "thing" which could be a supernova, a star or neutron star or black hole or something else. While the term "merger" might be used from time to time by some folks, "collision" will be understood by all.
Things smaller than stars (e.g. planets, protoplanets, asteroids, dust...)
I think that again in this case "collision" will be the right term, though in solar system formation there is plenty of merging of objects to make larger objects. This needs to be explored further as this answer identifies itself as a "Partial answer to share what I've found to date".