They honestly don’t really look at all. Show license with card just showing my name and they say you are good. It’s just ridiculous and a waste of time.
I can reconfirm if you have a picture of the front of your vaccine card on your phone, that's all you need to show. From my experience, they do check that the name on the vaccine card matches the name on your license.
Remember that you need to bring a picture ID along with your vaccine card or image. (This is not voting after all.) They would have denied my daughter admission had I not remembered that I had a picture of her school ID on my phone.