#2. Sorry for Indecision
You're taking votes on Indian or Chinese for lunch, and can actually hear a co-worker's brain creaking under the strain as they grapple with this Gordian enigma. Overloaded mental gears grind and tear, screeching an error message that means "This unit is not rated to take part in regular human activities" but sounds like "Oh God, sorry, hang on, wait, let's see ..." The amount of processing time required is inversely proportional to the importance of the issue and directly proportional to the number of hungry people now seriously considering butchering and cannibalizing the idiot.
"In the time it's taken him to draw a condiment flowchart, we could have barbecued him."
They're adding the apology to the crime, and every neuron spent spewing "sorry"s is another not employed to make the decision. And they clearly didn't have enough to resolve that imponderable conundrum in the first place. They couldn't give a less sincere apology without engraving it on a bullet. Apologies are meant to be about other people, but these fence posts simply cannot imagine the third option of "If I can't make a decision, maybe my own nonexistent preference is less important than the lives of everyone waiting for me."
"On the one hand, there's me. The other hand is also used to masturbate."
This person is mugging you, but instead of money, they're stealing time, which you can't make more of. And they're always the kind of people you could easily flatten if they went for your wallet. But misplaced politeness arms them to steal precious life from everyone they know.
If you find yourself about to apologize for taking too long to make a decision, say the first choice. Instantly. If they were so close, it doesn't really matter which you pick, and more importantly, you no longer deserve to get the best possible choice. That was a prize for people capable of making decisions. And if you're in charge of collecting the responses for any casual vote, just count "um"s as spoiled ballots. It doesn't mean "Pause everything and wait for me, the most important member of the group"; it means "Leave me, I'll only slow you down."
#1. Sorry Postscript
By far the worst and most common anti-apology is the postscript. You've choked out a "sorry," things have calmed down, and you can't help but sling one last barb. You might as well have spent the last five minutes yelling about the other party's sex with Satan, because the fight is back on, and you could have had some fun inventing new mythological matrimonial relations. An apology with a postscript is a box of chocolates that smells of urine: It doesn't just destroy the effort, it means they're more pissed off that you went to all that effort to do a good thing and then went to even more personal effort to ruin it.
"They say you should put a lot of yourself into a gift."
The idea of "winning" an argument is the exact opposite of apologizing, and also why apologies are necessary. You can convince the other person in a discussion. When things move to argument mode, your personality is just a barnacle clinging on to the back of a huge white whale of territorial instincts and posturing, and the whale only wants to kill that bastard Ahab. Once things escalate to the argument stage, you really are in WarGames: The only winning move is not to play.
This knows more about human emotions than you.
Get into a yelling match and you've handed over the controls of your modern brain to the monkey that lives one inch below your skin. And that monkey might now have access to an improved vocabulary and an impressive arsenal of psychological weapons, but its tactics haven't changed: screech loudly and fling shit at the opponent.
Every time you've ever yelled, except he has an excuse.
If you're going to do anything, do it properly. Either apologize and stop talking, or down a Scotch, smash the bottle, set fire to the table and leap out the window. Either way, they'll admire your honesty and know better than to keep arguing with you, and you'll get some peace.