Various ways of fixing chucks to lathes, each with pros and cons.
A single screw on spindle is easy to make, accurate, and reasonably quick-change but the chucks tend to get stuck and risk coming off if the lathe is run in reverse. So best for lathes that run in one direction only. Ask the forum if you ever need to remove a stuck screw-on chuck! Broken teeth on Myfords is a common fault because ill-advised owners use the back-gear to hold a stuck chuck so they can apply excessive force.
Bolting the chuck to a flanged spindle is easy to make, accurate, never sticks, and can run both forward or reverse. But changing the chuck is slower and fiddly, especially on a small lathe if you have big fingers.
Best of all are the various forms of cam-lock. These are accurate, never stick, do forward and reverse, and are quick-change. Not easy to make and hence expensive. Typically made for workshops where time is money.
Although chucks can be made to fit one specific lathe, it's usual to attach a standard chuck via a backplate that adapts it to whatever spindle fixing the lathe comes with. Makes it much easier to buy and attach chucks.
Of the cheap and cheerful systems, screw-on seems to have been most common in the past, whilst most modern machines favour bolt-on. I think this is because most modern lathes are designed to run in reverse which it's very convenient for some operations such as metric screw-cutting.
Dave