Reading over the Monk I saw that for Martial Arts:
When you use the Attack action with an unarmed strike or a monk weapon on your turn, you can make one unarmed strike as a bonus action.
This implies that non-monk weapons do not count towards being able to make this single unarmed strike. However, when looking over Flurry of Blows:
Immediately after you take the Attack action on your turn, you can spend 1 ki point to make two unarmed strikes as a bonus action.
The requirement doesn't seem to be there anymore, and I didn't see where this discrepancy was covered in this answer to a similar question about the limitations of an unarmed strike. This answer to a duplicate question says that you can use flurry of blows as a bonus action on any form of attack.
So does the limitation really only exist on a single unarmed strike, but you can use whatever attack action you want to trigger Flurry of Blows?