(I host a STUN and TURN relay myself, because I had to for my personal VoIP server for enough people to be able to connect on it. Downside is more use of bandwidth.)
edit: replaced STUN with TURN where appropriate, I did confuse both as they were provided as a single package.
STUN is not a relay, but TURN is, and STUN/TURN is a common combo for when STUN doesn't manage to holepunch reliably, falling back to the relay when the direct connection fails.
What's also true, and what I think the GP was trying to get at, is that STUN requires an external coordination server. UPnP (I think—I am far less familiar with it) does not, because in UPnP you're negotiating the holepunching with the local router directly, whereas STUN is sort of using a loophole.
With TURN, all the traffic to the clients is routed through the TURN server indeed. That makes hosting a discussions server more traffic-heavy than otherwise...
(and it turns out that the server software that I use implements TURN and STUN in the same daemon)
And TURN is one of those relays.
(I host a STUN and TURN relay myself, because I had to for my personal VoIP server for enough people to be able to connect on it. Downside is more use of bandwidth.)
edit: replaced STUN with TURN where appropriate, I did confuse both as they were provided as a single package.