For those of you with older/grown up children

[roseability]

Do your children remember the times that you made mistakes?

Or if you are basically a good enough parent and apologised for your mistakes have the good times had more impact than the bad?

I have had a bad week and I lost it with my ds (aged 4). I pushed him down on the sofa and really shouted at him

This does not happen often but I can't help but feel it will damage him in some way

Or maybe I am paranoid because I had a rotten childhood. I apologised unreservedly and we hugged afterwards but is that enough?

Yes, they remember. It teaches them that when they screw up, there is always the possibility of apologies, forgiveness and greater awareness. How to resolve conflicts and mend dented and battered relationships.
You need to develop some other strategies, walking off, deep breaths, whatever it takes to cool the heat of your initial reaction and give you time to respond calmly.
No, mine weren't scarred for life, yes they are fantastic teenagers. 4 Year olds are very trying!

What Goblin said.

They do remember but they dont bear grudges :)Mine are parents them selves now and we laugh about some stuff and I cry over some.The worst thing I ever did [when dh was ill and abusive and I couldnt bear it] was to get in the car at night and drive to a layby on top of a hill and look at the view of the village lights down below.I needed silence[dh wasnt physically abusive but emotional and verbal].My dc were about 7 and 11 and I always said,'Mummy's coming back' but apparently they didnt believe me and used to plan how they would manage with dh if I didnt.They used to sit at the bottom of the drive and watch for my car returning then they'd scurry back to bed.I only found out 25 years later.I hate myself for that.
My dd once said 'you did your best' but I dont think I did-I reacted to the moment rather than acting as a thoughtful and loving mother should.