Hi Joseph,
Welcome to planDisney! Your vacation is just a few days away, so I'm glad you dropped in with your question before you finish packing.
These days, we all have our favorite personal containers for keeping our water or soft drinks readily at hand. The good news is that, as long as it isn't glass, you are welcome to bring your personal water bottle or cup of any brand into the Disney parks. (You can bring snacks and non-alcoholic beverages as well. You'll find the
list of prohibited items and park rules here.)
I usually carry a smaller bottle without a handle that I can stow easily in my bag or backpack. But I've seen hundreds of cups like the one you prefer in stroller cupholders across the parks. It's a matter of personal preference. The important thing is to stay hydrated in the September heat of Central Florida.
You will find bottle filling stations scattered around the parks and resorts. And any quick service dining location will give you a complimentary cup of ice water, even if you don't purchase any food. I also try to remember to fill my personal water bottle before I leave a table service restaurant.
Thanks again for your question, Joseph. Please let us know if you have any others as you make your final preparations.
Enjoy,
Wilma