"Running water" doesn't simply mean water that is moving at the moment.
"Running water" as a term specifically refers to rivers and streams, not just water that happens to be in motion. A thrown water bottle is not 'running water', nor is a small continuous leak from a water pipe.
Today we more commonly use the term 'running water' to mean indoor plumbing, but whether a vampire can safely pass through an active sprinkler system or be harmed by a firehose is beyond the scope of the discussion here. In the context of a quasi-medieval world, 'running water' means something like a creek.
So in most cases, shape water probably can't produce anything that would be harmful to a vampire -- potentially, if you were close to a river, you could redirect some of the flow of water slightly beyond the banks to engulf a vampire (per the "change the flow of the water as you direct" clause), but it would be up to the DM to determine whether that counts or if you're just pulling a single lump of water out of the river, in which case that ball of water would not have a flow and would mean nothing more to the vampire than dumping out a bucket over its head. In either case, water out of a barrel would definitely not count as running water, even if you got it to move in the short term.
Historically, most of the vampiric weaknesses are things seen as sources of purity and cleansing -- sunlight, running water, salt, garlic, fire, and holy water were all associated with purifying the unclean or just being inherently pure. Running water in specific was often associated with the holy rite of baptism, in addition to being where you go to clean everything from your clothes to your body. (If garlic and salt seem strange on that list, remember that before refrigeration, salt was a preservative, and strong spices were often used to cover the taste of meat that was starting to go rancid. This was several centuries prior to germ theory, so covering up the taste and smell was seen as basically the same thing as actually purifying the food.)
D&D doesn't directly operate on historical intent, but it seems to me that knowing why this specific collection of relatively random things are meaningful to a vampire should give us insight into what does and doesn't count.