I would break the NCAA Tournament into the following eras:
1) Pre 1960 - Basketball was heavily segregated and the tournament was only 8 teams until the early 50's, then expanding to 22. The Tournament was heavily regionalized, so East played East, Midwest played Midwest, etc., until the Final 4. Also, the NIT was considered the more prestigious tournament to some schools so teams would occasionally choose the NIT over the NCAA. These championships basically don't count.
2) 1960-1974 - Integration came to college basketball, and the NCAA forbid teams to play in the NIT if they got an NCAA bid starting in 1970. There was still a "one team per conference" rule until 1975 and the NCAA Tournament was regional until the early 70's. This is how UCLA would play UC Santa Barbara and New Mexico State almost every year in the Sweet 16 and Elite 8. These Championships get an asterisk.
3) 1975-1984 - Near modern era. Tournament was expanding (32 teams in 1975, 40 in 1979), and allowed multiple teams per conference. There were still byes and some other weird stuff (regional teams got seeding preference through 1978), but it was getting closer to the modern tournament format. These championships basically count.
4) 1985-Present -Modern era. 64 team tournament starting in 1985, Shot clock starting in 85-86 season, and 3 point shot started in the 1987 tournament (although some conferences had been using it for league games since 1980). Other than the haircuts and some conditioning, the game that was played in the late 80's had fundamentally the same rules as the game played today. These championships DEFINITELY count.