Partial answer to share what I've found to date:

Galaxies

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:

• Milky Way Formation
• What parameters determine whether galaxies colliding will result in a merger or a hit and run?
• Why do we believe that the super massive black holes at the centers of two merging galaxies would themselves merge?
• Difference in energy released in stellar mass black hole merger and supermassive black hole merger
• Why can't supermassive black holes merge? (or can they?)
• How do two supermassive black holes reach "the last parsec" in the case of merging galaxies?
• Are there galaxies with 2 or more super massive black holes orbiting each other? "Yes, there are galaxies with two supermassive black holes in the center, see for instance 4C +37.11. Most likely such galaxies are formed by collision and merger of two galaxies, and their cores have not yet merged. Source"
• Is there a way to calculate how much damage black hole merger shockwaves inflict on nearby objects?
• What enhances the capture and merge rates of pairs of small black holes orbiting around supermassive black holes?
• Why would the merger of spinning black holes within the accretion disk of a supermassive black hole cause them to "shoot straight up" out of the disk?
• How do we know that supermassive black holes can gain mass by means other than merging with other supermassive black holes?

Stellar objects

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.

• How likely are planets to form after neutron star collisions?
• GW from merging of neutron stars and black holes "So in the end, the frequency of LIGO detections depends on two things: The real distribution of objects in binaries: NS-NS, NS-BH, and BH-BH and the loudness/brightness of the gravitational waves emitted by the merger."

As for stars merging:

• Could our Sun be the product of an ancient stellar collision? "Stellar mergers are certainly possible, but also relatively rare. [...] Glebbek's dissertation on stellar mergers estimates a rough condition for the orbital angular momentum to exceed the maximum spin angular momentum of the merged star as..."
• Could our Sun be the product of an ancient stellar collision? "The dynamics of the Solar System and the chemistry of the Solar System bodies don't support a hypothesis of a stellar merger later than formation of the protoplanetary disk" and "Stellar mergers are certainly possible"
• What would happen if a Sun-like star were to consume a Jupiter-like planet?
• How far from Betelgeuse is its habitable zone? "In addition to being a highly unstable and variable supergiant, it's a runaway star, suggesting that it was formerly a member of a multiple star system with a companion star that went supernova. Its relatively rapid rotation is difficult to explain via single star evolution, suggesting that it has undergone a stellar merger (Wheeler et al. 2017, Chatzopoulos et al. 2020)"

What can be said?

1. "Merger" is a solid, standard term when it comes to galaxies and their supermassive black holes.
2. "Merger" is becoming a standard term for the last few moments of NS-NS, NS-BH and BH-BH grabitational wave event.
3. 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".