Big mistake from myself and plenty to learn from, but
why does the manager feel the need to lie!? Gave the pen immediately, apologised at the end of the game...but otherwise it's spot on
Because you gave him a get out of jail free card! His team are near the bottom of the table and were playing one of the top teams with a scratch side. He can either say "we were well beaten by a better team!" or else "we tried really hard but a disgraceful decision robbed us. We were just getting back into it but the Ref cost us the game!" You gave him the chance to lay it on with a trowel and make it look like your fault and I'm afraid he grabbed it with both hands.
I had the same thing last week - although I'm no longer doing games with hundreds of spectators - and all because I don't use club assistants. I had a team moaning about an offside that "cost them the game!" A free kick into the box was partially cleared and most of the defence ran out, all apart from one defender who stayed in the area - right next to me - playing the attacker on by three yards as he collected a through ball and scored the winning goal. It doesn't matter that they were 3-0 down at half time and came back to 4-4 after I sent off an opponent for DOGSO-H just after half time, so they were unable to hold out against ten men while playing with a gale force wind and driving rain behind them. What cost them was that goal. The player who was slow out was the youngest player on the pitch and didn't have the experience to move out with the others. I didn't drop him in it, because he knew what he'd done and will learn from it. His team would have destroyed his confidence if I named him, and I certainly don't need help standing up to players so I just said one of your players was slow out and left it at that.
As I said to them, if I had a club assistant he would have been opposite me level with the second last defender so if he'd flagged for offside then I would have over-ruled him anyway because he would have been wrong.
Teams are very subjective with their recollections of incidents, and focus on apparent injustices. All you can do is to give it the way you see it.
As others have said, if you think you have genuinely made a mistake then you can change your mind, although it's best to speak to the AR to at least make it look like he's helped you out rather than you're going with player reactions. However, the write up says it was on the left side of the area so probably far side to the AR so not really credible, but it will allow you some thinking time at least. Also, the next time there's a contentious decision the players will ask you to speak to him again.
It's probably best to avoid saying things like "I was wrong" with "characters" like this. (I loved the woollen Benny hat and the "Ref, what are you on about" pose in the article.) It's better to say things like "from my position it was a definite pen!" and leave it at that. Players don't always appreciate an honest referee and you have given them ammunition to attack you with.
Try to delay a second to think about if you've actually seen what you thought you had. If there are appeals then it makes the decision credible. If there are no appeals then you may have been wrong and have a decision to make. I know we don't go with the shout - but player reactions do help us out at times! On this occasion they have done the opposite.