How to get a puppy to stop biting you when you grab them by the collar?

I would continue with the leash. Luring with food might be good to, to keep things positive.

It's really hard to comment without knowing more details, like what frightened her. When I hear of a 1 yo biting, I think the dog probably dominates the owner, hasn't been trained, and is only going to get worse... But, the way the owner described her (to you), it could be the dog is only obedient to people it respects as a pack leader. It may not have reached the same level with you that it has with its owner.

The best way to deal with that is as you're doing: use a leash. Be formal. I do that with most dogs I board, until I trust them with more freedom. The first hour or two. It also helps establish that position that she might see in her owner, but not in you. Some dogs are hard to impress. You might have some tall shoes to fill (in the dog's eyes).

I agree with everything Mark says. If you have stay with this dog again, I'd ask what words are used for commands. Sounds simple, I know, but many clients and many trainers have a different verbiage for common commands. For example, my last dog who was a professionally trained hunting dog responded to "HUP" instead of "SIT". My whole life I always trained my own dogs to "SIT", and then come to find that it's common for hunting dogs to sit when told to "hup". "DOWN" AND "OFF" seems to be used interchangeably as well.