Hi. We had our wedding reception at Plunketts (it was just a sit-down dinner with about 20 people, it's a regular restaurant, not a wedding place). The food and service were lovely.
Russells Restaurant has nice traditional English food.
If you like Mexican food, you can try Fiesta Mehicana on Clifford Street.
I have never been to Betty's, but it always looks packed.
FYI, parts of York city centre are pedestrian only, so be prepared to do some walking.