My opinion on different types of crossovers
Crossover: Not sure if it's the most common but it feels the most prevalent. Usually for characters with drastic settings. Send character(s) through a portal for a sort of meeting of the minds. A fun part is exploring differences. Tv shows and movies usually have a way back, comics as well. Extra angst if there's loved ones left behind.
Semi crossover: How shared universes usually go. The settings are similar enough it's not impossible that they were in the same universe. Could vaguely be aware of other events for crossover convenience. Very common for tv shows, movies and comics. Lets characters interact without multiversal implications.
Soft crossover: Sometimes settings are too drastic, but writers still want characters to interact without multiversal implications. Explained through variants with a reworked in universe backstory. A character's canon is shaped by the setting it takes place in. Fun for alternate universes. Comics and tv shows might do this with cameos or parodies.