This probably comes back to what part of the context is left unsaid. If someone's favorite mug is broken, you are trying to convey something like: "I'm sorry (to hear that)" But an apology obviously has the meaning of : "I'm sorry (for what I did that made this happen)"

Being explicit about it could help, but I agree my default inclination is also to assume it's an apology.

More specifically, in addition to remorse for one's actions, 'sorry' indicates you're grieved over a situation in general. So, if somebody's favorite mug is broken and you say, "I'm sorry" you're indicating the state of their mug grieves you.

(Grieves might be a bit heavy of a word to use, so saddened works too—the concept is the same.)

In the UK a common response on learning of a bereavement or some other major life setback is to say "I'm sorry" or "I'm so sorry."

Obviously this doesn't imply guilt.

If your most recent code breaks the company website, the correct response is probably "I'm on it - ETA of <x> mins/hours."

You could argue that an apology is redundant, although a post-mortem may be useful.

It's a difficult cultural issue, because it depends how avoidable show-stopper problems are.

The law doesn't fully distinguish between negligence caused by personal error; negligence caused by stress, distraction, or overwork; trivial error which may be magnified because of circumstances or other contributory factors; basic incompetence of one individual; basic incompetence of one or more other individuals which ends up being the problem of someone who isn't directly responsible (e.g. a site crashes because of an ancient bug buried in ancient code); and a series of extremely unlikely but unfortunate events.

I think the word's best definition is as an expression of regret rather than repentance; it's probably clearer to most that regret isn't necessarily an admission of wrongdoing, nor even a feeling about one's own actions.