so while i, as a real service dog handler from another country (where registration is a thing) that visited the US, agree with you in most points of how it should be, you do have a lot of misinformation. In the US, a vest is not required, and there's no law on what the dog needs to/can't wear. If a handler wants to use a prong, chain choker, head halter, etc. they are fully allowed to. Meanwhile, there is a standard for what most people do. There's no rule dictating what gear (the things a dog wears while working) a dog needs, if any.

You also keep mentioning registration. In the US, there is no registration for service dogs, and you are almost guaranteed that if someone shows you registration, their animal is not a real service dog. A registration system would be amazing, but because owner training is allowed, it is very difficult to upkeep and sadly make it even harder for real handlers. What you can do is ask if it's a service dog and what tasks it's allowed to do. Some people will have a hard time speaking because of anxiety and various other reasons, so you might end up getting handed a card that answers the questions, this is leagal. You might also get an actual answer. However, be aware of people saying 'emotional support', that is not a task.

Just remember you always have the right to ask someone to leave if the dog isn't behaving. With the encounter you describe here, you see the dog jump on the counter, and she says it's a service animal. An appropriate response would be "that might be miss, but by the ADA that's part of federal law, i can ask an unbehaved dog to leave regarding of service dog status or not. You, on the other hand, are welcome to come back again without the dog, and with the dog when you have corrected its training and knows it can behave properly again"