You're contradicting yourself in some ways.
If you got too close to a man at work and you kissed it, it is an affair. Even if you didn't have sex with him. And TBH, if you were caught rather than coming clean, I don't blame your husband and other people, including your child, for thinking it was a full blown affair. But it doesn't really matter if you had sex or not. You still cheated and got caught. Not sure what you want to say to defend yourself.
Your DD may not be happy at all that you're minimising your affair and your betrayal of the family.
You said you had no other choice but to go and see your DD without telling her, so don't claim that it was a casual encounter when going home to pick up things. Even if it's your house, you don't live there and you're not welcome there. In your ex's place I wouldn't be happy for you to just walk in when you please. You must know when people are likely to be at home or not, or you could (should, really) tell them when you're going there.
For the important part, your DD got no warning and people can fight, flee or freeze in uncomfortable situations. It looks like she froze and didn't know exactly what to do or say when suddenly meeting you. So, she went along with it. She clearly had second thoughts about it and probably got angry with you for ambushing her. I would.
Don't do anything like that again.
She probably fells you've betrayed her, just as much as her dad. You must have caused her much hurt, in addition to the hurt she has seen her dad go through.
So, give it time. Let it heal. Keep telling (and showing) her you love her. Be honest with her. Respect her and her boundaries.