The specific scenarios you list are up to a DM
Running water isn't actually defined as a game term, so we have to go by general definitions. While that can be defined, the specific scenarios you ask about are really only answerable by your DM.
If you are the DM, then you need to make a reasonable ruling on them.
How I'd rule
For me, running water is either a naturally or magically occurring event. I'd try and keep an open mind for cleverness, but to answer your specific scenarios:
Rain is not running water. It's falling water. I'm also not familiar with vampires not being able to be in the rain. Dark and stormy nights and so on.
A bottle is not running water. You're pouring water out.
Urination is probably not. Clever, but it's likely to create more problems then it'll solve.
Puddles are not running water. They're literally standing water.