If I were to use a reusable water bottle should I be concerned about using one when I eat dairy and another when I have meat, or can I just wash it out?
Shulchan Aruch Yoreh Deah 88:1-2 (excerpts)
"Even with meat of a wild animal and a bird it is forbidden to put them on a table where one is eating cheese, so that he doesn't come to eat them together. But [to put them] on a table that's meant for arranging cooked foods, it is permitted to put one next to the other."
REMA (discussing when two friends dine together, one eating meat and the other cheese)..."And they should be careful not to drink from a single vessel because food sticks to the vessel. And of course they should not eat from the same bread. And so is our custom to have individual salt-vessels for each person themselves, because sometimes when they dip into the salt there remains leftovers of the food in the salt."
The Rema explains that we should not eat cheese, and then drink from a cup, afterwards passing the cup to a friend who is eating meat, so he can use the recently used cup. (or vice versa)
So, since these two diners are willing to share "germs" and "cooties" :) , we do fear that meat residue on a recently used cup will be licked/eaten by the second drinker who currently has a mouth full of cheese, at the same time.
From here we can see that a person alone, using a water bottle while munching cheese, should not soon afterwards eat meat while drinking from that same bottle. After all, you are your own best friend. :)
The Shach says that such a mixing of use is in fact "forbidden".
The Kaf HaChaim however brings down that it is "incorrect or inappropriate" behavior.
He then goes on to say that if you and your friend (and therefore you alone as well) are careful to clean your mouth before using the cup, or wipe your lips well before using the cup, then its OK.
No authority that I know of would go so far as to say the vessel becomes a milk vessel or meat vessel etc. simply because you drank cold water from it at a meal with cheese or meat.
Of course, if you do see with your eyes, a blob of cheesy or meaty substance floating in the bottle, then it should be washed out before using at the opposite meal.
The problem seems to be a real residue of actual foodstuff clinging to the mouth of the vessel.
Backwash from your throat, dragging semi-digested particle back into the bottle, doesn't seem to be a concern here.
It would therefore seem that washing the bottle out and cleaning the rim would definitely be more than enough, (and count as an an extra effort to beautify Kashrus observance,) to use it for a later meal of opposite food.
Cleaning the rim by itself, without washing it out,(and of course making sure your mouth is now clean from previous food) does seem quite adequate without the need for changing the water.
So if you drink half a bottle, and then inspect the rim and wipe it, there should be no need to dump the other half of the bottle if you need to save that water for a later opposite meal.
Of Course, its always nice to have two bottles, because anything that's done to enhance separation of milk and meat is usually a bonus in Kashrus. Some also consider it nice to clean the bottle out and dump the water, if water is usually not scarce or expensive.
NOTE: This applies to a bottle that is used casually for water etc. A bottle used for milk, or gravy, or one that is absorbing a lot of backwash residue and particles of meat and milk, (especially if made of pottery) or under conditions of heat etc. is a different issue which is beyond the scope of the OP.
As always, these are the Halachic sources. For a practical decision in a particular case, you should always ask your local Orthodox Rabbi.
I hope this helps. :)