There are no definitive lists of medals that I am aware of, but it seems that stuff is coming in-game with Forsaken.

Is there a complete definitive list of PvP medals with their descriptions? I’ve only come across incomplete ones, which just happen not to have the ones I want to look up and find out what they mean. Kind of surprised this isn’t in the companion app.

As Speed says, there are no official lists, and names vary quite a bit (I guess the API doesn't match what we see in-game), but I've found DestinyKD.com to be pretty complete in its descriptions. Find the game that contained the medal you're curious about, click on the gamertag of the Guardian that won it (you, probably), and there'll be a list of medals, with descriptions of each.

Not super-efficient, but accurate, at least in my experience. Downside: If the game is old, I'm not sure how to find it.

This late in the game, I doubt even Bungie knows. Remember, they didn't even have player stats on their site until what, last year? Even then, they only show the number of kills and the final scores. It honestly feels like they were doing it out of spite-- Once they saw people create the amazing DIM with their API and other useful sites, it was basically like they said "Fine, we'll let you handle it." The handful of games I played in Crucible during the Beta and launch game, I only understood about 20% of the medals they had on there, and most of those were from the last game.

Another thing that degrades over time, becoming worse instead of better. Remember the old Halo stat page? Those damn things had literal markers that showed exactly who killed who, in exactly which place and exactly with what. Now you can't even find out how many kills you got in a game unless you take a picture of the carnage report. Sad day.

I guess Bungie updated the API with medals and descriptions in prep for Forsaken.

https://docs.google.com/spreadsheets/d/1YL_4oNzW41HB8Oes60IPmmFR1mr9FR7yMezEJSLcOZM/edit#gid=0

Someone compiled them all.