No, the 10/22 magnum will not fire .22LR.
The problem with the .22 magnum is that you don't gain much over the .22LR. The ammo is harder to get and more expensive and the rifle itself is almost twice the price of the standard 10/22. The manufacturers know that the 10/22 is the most common .22 rifle (hell, maybe the most commonly owned rifle PERIOD) in the country, so they make their products available for it knowing there is an excellent market that isn't going away. The 10/22 magnums are relatively rare. Just try to find a used one online for sale and you can get an idea of how uncommon they are.
If someone is going to step up from .22LR, usually they go with a .17HMR or something centerfire.
You say regular 10/22 and then 10/22LR... there is no "regular 10/22". The 10/22 is .22LR. I'm not sure what you were trying to say there.