You say you don't act differently, that you focused all your attention on dd and yet you skype this man all day everyday
Even my 5yo would notice mama spending more time on her phone/laptop than with him!
You may have known this man for 6 months, but you introduced/told your dd 3 months later, that's a very short amount of time for her, and is moving too fast for her.
A 13yo knows concept of time. Do not belittle or partronise her dd making her views on the timeframe non important and silly.
I agree that your dd cannot dictate your life or relationship and needs a few rules in place but she is allowed her opinion. You can see this man but do not bring into your daughter's home when she has made it quite clear it makes her uncomfortable and doesn't like. A home is supposed to be a safe haven for a child.
You are right, age does not matter nor are we to be judgemental of that but in the eyes of a 13yo, 54yo is a huge difference and especially if he looks older than he is. She may evenn be embarrassed of the age gap between you too.
The way you phrased "stayed for their sakes" souned at lot blame and resentment.
6 months together and you are already thinking of bringing him into the family dynamic in future? RED FLAG. It may not be fast for you but it is for dd.
She is vunerable and insercure. She needs her mum!
You do not want a compromise, you want a quick fix so that you can play happy families. Your daughter is not ready for a new male figure in her life. Respect that, for her sake.