20" is the maximum OUTSIDE diameter of the tyre once fitted
Narrow 1-1/8" & 1-3/8" rims use different tyres but are still only 20" max when inflated.
I could be wrong but I think it has something to do with very early racing rules but either way there are two sizes, one is denoted with a fraction in the notation and the other uses a decimal. A standard rim is termed 20x1.75" or 20x1.5" and will fit any tyre from 20x1.5" upwards, seen newschool running 20x2.4" fairly often. The other rim is noted 20x1-1/8" and will run 20x1-1/8" or 20-3/8" tyres
Hope that helps - can probably post a few pictures if it's still not clear but for example my kids race bikes have a 1-1/8" narrow rear tyre and slightly wider 1-3/8" front for better grip
Oh, and don't forget you will need different size brake caliper - probably 610 size not standard 890/900/1000/1020's - if you're not sure what I mean the measure the centre of the brake spindle hole to the middle of the rim brake surface in mm then add a zero which will give you a number to aim for in the serial size of the caliper