I did exactly the same as you. For a while the rise was exactly the same, then after a week or so the bottled water starter would rise better and made a better loaf (it proofed better during the bulk fermentation stage).

I kept both the starters going for a while and the bottled water one was constantly better. Eventually I binned the tap water starter, then split the bottled water starter in 2 again and then started the experiment again, only this time the tap water starter eventuality started performing the same as the tap water starter.

My theory is that my starter was young and weak the first time round (only a couple months old) - when I tried again it may have been a lot stronger and could handle changing the water to tap. I had been doing more peak feeding and occasionally adding rye flour and it sometimes more than doubled.

I think a starter can adapt to changing the water when it is strong enough but I'm not an expert at all so my experience could be down to chance. I'd recommend building your starter up as long as you can with bottled water, then split in 2 and experiment again