If my neighbor suddenly stops talking to me..(true story here) and I, after several days of silence walk up and ask if I have done something to offend you..and I get stink eye and more silence as they walk away, And I say.. "I apologize if I have upset or offended you"... I am being insincere, or insecure? We never had any problems, and suddenly they stop talking to me..
I was not being either..I was attempting to make peace with a neighbor..over something.. I still don't even know what..is it my problem anymore or theirs?

My mother gets offended if someone swears and she hears it. A stranger drops the f bomb and my mom turns and says to them.."Please don't talk that way around me"...the stranger gets a weird look and apologizes. Also a true story..
was the stranger being insincere or insecure? I think just trying to be nice..even though maybe he felt my mom was being intrusive..which she was.
The comment made by Gator.."that if you think I did something wrong, I'm sorry you think that."..I don't feel is an indication of insecurity or insincerity...who knows what people are going to be offended by.
I have no problem saying I'm sorry...I can not control what others feel is offensive and apology worthy.
I wouldn't want an insincere apology either..if one feels the need to apologize to me I will accept it..I never demand or expect it. I get to choose what offends me..and it ain't much.

I never said I want to. I Just wanted to apologize for not being the president.

dont apologize,,,its a complicated job which most of us were never meant to do and probably not equipped to handle.

That's an apologist statement. "never meant to do"..."not equipped to handle"...I can handle and do everything I want to.

I guess my point was that my experience is that when people say..."that if you think I did something wrong, I'm sorry you think that."...it's not really an apology...or a left handed one at best. They're saying you're wrong...I didn't do anything wrong. And, again, MY experience is that these people are almost always insecure and/or insincere.

I know that..we mostly go on our own experiences..that is why I posted what I did. In some cases you are right..is just one of those things that isn't absolute. There are times when some do just get offended and one is kinda left wondering ..what did I do..and so they seem to stumble through an apology to attempt at not hurting feelings...to intentionally insult and hurt...then apologize, agreed, who needs it. I like to think most people are not so unkind...but that's just how I look at most people, basically good.

so call me an apologist, I dont believe everyone has the same resources and therefore not the same capabilities.

How can you be sorry for something you don't know what you did? I would not consider it to be sincere.

yeah , that was my point to. Id rather one apologize that they caused me pain instead of claim to be apologizing for something they dont have knowledge of.

Of course not.

I think that to actually be sorry for something and to admit and tell the person that is offended face to face, takes guts and when I have done something wrong, telling the person I am sorry is my obligation to making things right, it's not always easy to say "I'm sorry."

I think it can be sincere, because in these cases the offending party in general doesn't want to have offended the other person, it wasn't an intentional offense to the person otherwise held in some sense of regard.

To try to make amends, to repair or set right any situation begins with an awareness of having offended someone and apology, followed by looking at oneself to determine why I did that, my part in it, and an amend if required.