You should be fine.
A long time ago in France, wheels were measured by the diameter in mm over the tire and you had size 700, 650, all the way down to 400 on kids bike.
There were also different widths, so they went 700A through 700D from narrowest to widest.
This system is great if you are trying to figure out if your fenders are going to fit around the tire, but terrible if you are trying to match tires against rims, because a skinny 700A is going to not be very tall and thus have a larger diameter rim than the balloony 700D. And rim diameter is what matters most for tire compatibility.
Over time most of the other sizes went extinct except for the 700C and 650B. Other countries all had their own ways of measuring.
The real way to tell if it matches is to measure the diameter where the tire meets the rim. That is your 622 number.
Most shops will be calling it by the 700C, not the 622, but if you get them is will say 700C in ink, but 622 in the raised letters. Don't worry. That rim diameter is extremely popular.
The smaller number is the tire width. That's less important and everyone measures it a bitt differently.