Is the mushroom plug done at a tire shop different than the Stop & Go plugger?
Yes - big difference. A car tire plug is usually about an inch or two across and is pulled in and glued in (or heated) from the inside. These will usually work for the life of the tire and will require a balancing after they are installed. The S&G plugs are just a mushroom shape and stay in by compression. There has been a wide range of discussions about those and most people don't like them. They sound like a good idea when you read about them. I don't like them. I have always used the string kits (red) with a little glue to help it slide in. If it requires more than two strings then it gets replaced as soon as possible because that is getting to be a pretty big hole. Sometimes a traditional patch/plug can fix this. Never had any issues other than a few slow leaks over the years.
Disclaimer: If you aren't experienced in your own tire repairs - do not attempt. Only repair the tread area (up to about 1" from the edge). Never repair a sidewall hole unless it is to get you home.