Alright, so after getting another Gold present in a live I played and counting the number of logos, I'd say it is safe to assume that the number of notes that must have logos is ~45% of the total notes in a song. I had 95 appear out of 207 notes, which is the max chain for edeN on Pro difficulty. This is actually slightly less than 47%, but the two previous instances showed about 44%, so averaging it out would make it about 45%.
So Normal presents can be received(but are not always guaranteed)with anywhere from 1-5 total logos throughout a whole song, Silver presents are earned when ~33% of notes have logos, and according to the three times I was able to use as examples, ~45% of notes having logos gives a Gold present.
I am sort of in the process of testing which all difficulties that Gold presents can be received in, but not too seriously as it takes time even for me; if you know, then you know :p and I'm not sure if anyone only ever plays on Regular/Debut difficulty.