Like so many English phrases, “I’m sorry” sounds simple, but it can mean a lot of separate things – all related, but subtly different.
I can think of at least five meanings for “I’m sorry.”
Sympathy for event I didn’t cause. As in, “I’m sorry your pet marmoset is going prematurely bald.” I’m not apologizing, because I had nothing to do with your monkey’s thinning hair. I just want you to know that I feel your pain.
Sympathy for event I caused out of necessity. As in, “I’m sorry I couldn’t go to your Arbor Day party, but my basement was flooding.” I did something that might’ve hurt you, but I still think it was the right decision, and if I could do it over, I’d do it the same way again. This is bordering on an apology, but there’s no real regret.
Sympathy for event I caused out of perceived necessity. As in, “I’m sorry I elbowed your nephew in the solar plexus, but it looked like he was reaching for his Glock.” I did something that might’ve hurt you, but I still think it was the right decision based on what I knew at the time. If I could do it over, I’d do it differently, but only with the benefit of hindsight. My emotional brain might regret what happened, but my rational brain says there’s no realistic way I could have handled it better.
Sympathy for event I caused accidentally. As in, “I’m sorry I dropped your nineteenth-century commemorative Louisiana Purchase porcelain Christmas ornament on the floor and it shattered into 87 pieces – I didn’t mean to.” Most people would call this a real apology, but I’m still not into the territory of genuine fault (unless I was being careless), because the event happened outside of my ability to control it. If I could do it over, I’d try to do it differently, but if I succeeded, it’d only be with the benefit of hindsight. And, by definition, I didn’t have that at the time.
Sympathy for event I caused deliberately. As in, “I’m sorry I tore off the head of your Kylo Ren action figure and hurled it into the garbage disposal – I was having a bad day.” Here, for the first time, I’m admitting to deliberately doing something I knew was wrong: a full and true apology.
Of course you could think of other scenarios that don’t fit neatly into those five categories, and you could debate how much fault is required to warrant a “true” apology, or what that even means. As usual, the more precisely you try to define a word, the fuzzier it gets.
My point is simply that “I’m sorry” is a remarkably vague statement. It can refer to a wide range of situations with very different ethical implications. And since we rarely know (or even consider) which kind of “sorry” we’re talking about, it’s hard to know what to make of someone else’s apology, or when to apologize yourself.
You can always decide not to worry about it and just say “I’m sorry” whenever it seems to feel right, but if you do it too much, the words can lose their meaning and become mere reflex.
And what should you do if you reflexively say “I’m sorry” so much that it loses its meaning?
Well, I suppose you’d better apologize.
Brian D. Buckley 
There is one that is subtly but distinctively different and fits between the last two. When you do something that you should have known was wrong but just weren’t using your head. Like driving drunk. Or not calling your mother on mothers day.
Yeah. It’s strange, we like to think of guilt or culpability as a binary thing – it’s either your fault, or it isn’t – but there’s a whole wide spectrum packed into the concept. Which, I guess, is part of the reason so many Supreme Court cases are so difficult to untangle.
Here’s two more:
“I’m sorry?” (As a question, usually meaning, “Can you please repeat the last thing you said?”)
And the classic usage by politicians, as in, “I’m sorry if anybody was offended that I referred to my esteemed opponent as a ‘nappy-headed ho.'”
Meaning, I’m sorry this has become such a big deal on social media.
The “please repeat” one is interesting. I wonder if it originally began as a sort of apology, as in “I’m sorry [that I didn’t hear what you said]” and evolved from there. By now, it’s come so far that people can say “I’m sorry?” as an anti-apology, as in, “You’d better not have said what I think you just said, ’cause if you did, you owe me an apology.”
Gaelic uses a possessive rather than descriptive structure for emotions; e.g. I have happiness.
So, would use “I have sorrow” instead of “I’m sorry”; which fits all of your cases without creating the nebulous Venn of fault/non-fault/apology.
I’m afraid I don’t see the distinction.
In addition to expressing sympathy for sadness, sorry can mean apologetic or worthless; whereas sorrow only expresses grief and sadness.
Therefore possessing sorrow captures the same pain of mind that something has happened without the layer of potential moral judgement.