Sajnálom is another way to say I'm sorry. If you take it literally, the meaning is close to "I regret it" (referring to a bad thing that happened), and you can use it to express sympathy for something as well as apologizing for something you've done. Elnézést is strictly an apology - closer to "pardon me" or "excuse me" than it is to "I'm sorry." (If you want to break it down to a literal meaning, it's like "[I ask your] looking-away," although I don't think anybody really hears it like that.)

That is incorrect indeed. Mirror translation does not work here.
I can ask for forgiveness (Elnézést kérek.) or I can express that I feel sorry (Sajnálom.). Also see jsiehler's and vvsey's comment above.