622-17 is the size of the rim. The 17 is the rim's mounting width.

If you want to go faster, tire size is way, way down the list of things to change.

Legs and lungs are at the top of the list. Work on those and you won't need to fret over tires.

As for what fits on your bike, fietsbob is right. Measure. You don't need calipers. A cheap plastic ruler with millimeters on one side will do.