This is a... tough question to answer. Do I apologise? Yes. Is it a sincere apology for wasting time? Not really.

I can explain better with examples.

"Sorry for wasting your time with this mistake," feels very formal and isn't really applicable in most cases. It would be more appropriate if, say, I'd made a mistake, not owned up to it, and the lack of accountability has a negative impact.

"Ah shit, that's totally broken. Sorry, that was my bad!" is more what I'd do and it's a lot less formal. I'm not really apologising or feeling any remorse. I didn't do anything immoral, here. What I'm really saying with "sorry" is "I am accountable for making this mistake and I will commit to fixing it."

I hope this helps clarify the different meanings.