I've also been trying to introduce Erica to a sippy cup for about a month now.. I'd do either cooled boiled water or your PUR filter. Usually now that she is a little older I give her water from our filter too.
One thing I can def recommend tho is trying the BornFree sippy cups! We use Dr. Brown and BornFree bottles, but I figured since it was a sippy cup, it didn't matter as much
dohh
and tried her on the Playtex sippy cups with the hard spout. She kinda got the hang of it, and would try for a couple mins at a time, but never really was comfortable with it. I bought some BornFree cups, and she does great! The spout is soft and she likes that much better. She drank a whole cup full of water in it yesterday while on our walk. She even holds it herself!!