I found in a reference book that "I'm terrible = I'm sorry"
Does it make sense? I think it's nonsense.
"I was drunk last night, and I accidentally swallowed your goldfish. I know, I'm terrible!" With the right attitude of contrition, this might pass for an apology.
Indeed. 'I'm terrible' doesn't necessarily mean 'I'm sorry'.
It's more the feeling of being terrible. You can use it when you're being sarcastic too!
'I couldn't be bothered to arrive on time today.....I know I'm terrible!'
This also sounds a bit snobbish.......
In BrE, it's common to say that you are terribly sorry.