@beelola
at home, you must be having her run around naked... that is why she is can wee and poo on the floor. As NuffSaidSam said, you also must have put a nappy on her when she went out. The poor girl is confused, and only going to the potty when you suggest it, not from the signals that she gets. As a result, she is being trained to wee on command - not ideal.
Steps and Reasons
1 - Go to potty, sit and wee / poo based on her own signals, not your reminders Why so she learns to do same from the signals she already knows. Most babies hide before they poo - you want DD to go to the potty and poo.
2 - Pull down pants, sit on potty and wee / poo. By this stage, she knows the signals, now needs to learn the extra bit.
Some people wait for the child to be ready - i.e. the child informs the parent before they wee / poo their nappy. Since DD is pee /poo on the floor, she really is not ready OR is ignoring the signals that tells her that she needs to pee/poo
Going Out
Since she has pull-up training pants on at home, and will have learnt to tell you before she needs to wee/poo, she will be doing the same outside / overnight. As a result, contrary to NuffSaidSam, once out of nappies into training pants, one does not go back, even for sleep time.
The only exception is if DD is not getting this after 3/4 days.
In relation to DD loves having a wee on the potty, this is a new experience for her where she has your full attention. She will try and do what keeps that attention but if she does not have the focus to know prior to voiding that she needs to void, this is going to frustrate both of you. As a result, I suggest that you wait until she KNOWS.