I honestly don’t know if ‘concierge medicine’ is a different thing, but from what it sounds like and my own experience with direct primary care they sound equivalent. Possibly different by cost or target demographic.
We have DPC, and it's amazing. We pay $250/month for a family of 4. Some places charge less, but then also charge you for each time you see the doctor. Ours is all inclusive except for any in office incidentals - like lab work.
It's truly amazing. Our doctor knows us all well, can get responses to email or text within just a few minutes, generally same day appointments, or next day if she's really busy. Most things we don't even need to go in for - I did a recent international trip, and she just ordered me all of the needed travel medications without having to stop in.
A rose by any other name, and all that.