There are two distinct differences between those definitions.
First: for the limit definition, x is not allowed to reach c. That is, you have
For the continuity definition, you relax the condition so that x can actually equal c:
Second: as you've already pointed out, with the limit definition, no attempt is made to provide any indication as to what the limit is. It's just an unknown number L. Whereas, in the continuity definition, it must be f(c).