The NCAA DI Football National Champion is the champion of the DI-FCS subdivision. In fact, the NCAA doesn't even recognize the DI-FBS champion as a "NCAA Champion." Hence why the FCS team receives the standard NCAA trophy and the FBS team receives a different trophy.

As far as Dallas Baptist goes, it is a DII school that plays DI baseball. Similar to how Johns Hopkins is a DIII school that plays DI lacrosse. Ice hockey is another sport that has several DII and DIII schools competing at the DI level. During the early 2000s, the NCAA DI passed a bylaw that stated a school can't participate in basketball and football (FBS or FCS) unless it is a DI school. The bylaw also states that a DI school can't participate at the DII or DIII levels in any sports.
[Post edited by VTHokie2000 at 01/31/2020 5:18PM]