Download fixed for Trainer #6. Halo: The Master Chief Collection Trainer (CE ANNIVERSARY 04.22.2021), it now give the proper one. Downloaded and ran it myself to check.
I can confirm that the link on the page for the Halo: CE Anniversary version downloads the correct trainer, thank you for that!
However, the trainer itself shows that an update is needed and the Trainer Manager 2.0.8.0 wants to overwrite the correct trainer with the one for Halo Reach.
Since I don't know how the version check is handled (checksum on the server?) I can only guess that the info that is compared between the trainer itself/the Trainer Manager doesn't match with what's on the server.
It's not really a big deal since I can ignore the warning in the trainer and disable the check in the Trainer Manager, I just wanted to point it out to you guys.