I'm probably not the best to reply, because I have fostered god knows how many dogs and puppies over the past (more than a...) decade, so my dogs just meet foster puppies in the house, if the pups are reasonably healthy in appearances.
Adolescent dogs (6 months to a year, give or take, depending on the dogs behavior) that are bigger but still very "puppish" I typically introduce in our backyard or frontyard, leashed but relaxed. Again, my dogs are really good at intro's because they've done it so much.
With adults it REALLY depends, but as a rule of thumb I do not introduce adult fosters in a leashed, nose to nose, manner. I start with crating, letting mutual sniffing go on, then leashed walks together, etc. and move on from there depending on how long the foster will be here. If they won't be here long, I often don't really introduce them to my pack, or all members of my pack, at all. I only mention the adult - adult intro for others who may be reading and wondering about adult to adult intros vs puppy to adult intros.
IF I had an adult who hadn't spent much if any time around puppies, I would personally introduce them like I do adult - adult. Puppy in crate, let adult sniff. IF all is well and calm, then puppy would probably be in an x-pen and the adult could sniff and see pup bouncing around, etc. Make sure prey drive doesn't kick in here for some adults. I'd progress from there to either the yard with both leashed and different handlers, or if the pup walked ok on a leash I'd take them for short walks together for a few days.