In the opening post, you say you purchased the ticket for your friend because you felt guilty she had to go home.
In "your friend's email", your friend says they purchased the ticket and made an error.
Which is it?
If you want advice on whether or not to send something then you can't be asking people here to okay something which does not fit with the facts you have laid out. You should be truthful here, because if you aren't, the advice isn't going to be very helpful.
I don't think saying you accidentally added the railcard is going to be very productive - this is pretty much impossible to do by accident and there is a separate menu on the ticket machines to add it, and you need to press more than one button to select the railcard, especially if your explanation is that you did it in a hurry. So leave that out, nobody will believe that.