Thanks to MidAtlanticDad for tagging me earlier in this discussion. Sorry that I didn't have time to comment until now. I'm sure this discussion is pretty complete by now. D1 baseball is the only division in which a mid-year transfer as an undergrad can't be immediately eligible. (It is possible for a mid-year GRAD transfer to be immediately eligible at a new school in the Spring.)
Dad was also correct in that almost nothing stops the Division I 5-year clock as it's ticking down. Only exceptions are military service, an official religious mission, or there's a pregnancy exception for female athletes. Therefore, if a baseball player is cut in the Fall or is told that they are redshirting, the bottom line is that they won't be using one of their four seasons of playing eligibility, but their clock will keep on ticking whether they are redshirting at their current school or have been cut and are sitting out in the Spring.
Keep in mind that levels below D1 don't use the 5-year clock but instead have a 10-semester/15-quarter rule. That's more beneficial for Fall sports than for Spring sports in most situations, however.