The Torah prohibited labor done in a normal way. Violating that, someone would have to bring a sacrifice. (Thus, "liable.") If someone did it in an abnormal way, they are thus "exempt" from the sacrifice.
The Rabbis (about 2200 years ago, very loosely) realized that if people thought hooray, I'll spend all of Sabbath with loopholes, doing all sorts of labor in abnormal fashions!, then the spirit of the day would be lost, and what's more, people would eventually do labor the normal way too. Therefore they prohibited labor in an abnormal fashion.
If you spend some more time with Maimonides' code, you'll see there are actually three levels:
- "Liable." - Prohibited by the Torah.
- "Exempt." - Not prohibited by the Torah, so no sacrifice is incurred for violation, but the Rabbis prohibited it.
- "Permitted" - Actually allowed.
(If you hop over to his laws on divorce, he uses a similar shorthand. The divorce ritual could be void, meaning it doesn't even meet the Torah's definition of a proper divorce; or merely flawed, that it didn't meet the higher standard imposed by the Rabbis.)
So no, don't go about writing checks with your non-dominant hand.
Where it does come up is a situation where someone has to write because of a matter of life and death -- say you show up at the emergency room, and they demand you sign some paperwork. (They don't care how sloppily you sign.) Then signing with your non-dominant hand would be vastly preferable to doing so with your dominant hand, as the prohibition is a lesser one.
A famous rabbi's son was preparing to be drafted into Czarist Russia's army. His father had him study Maimonides' Code, precisely because it helps delineate between the minor and major violations. There was no way the fellow would be able to keep "normal civilian Sabbath" in the army, but this way he'd be able to keep his violations to the minimum.