One source of fresh drinking water for your cat may be your faucet. Cats instinctively prefer running water to standing water. It could also be that faucet water simply tastes better or is at a more preferred temperature. Faucet water may taste better when your cat's drinking bowl isn't cleaned and refilled with fresh water as often as your cat wants or the bowl is affecting the taste of the water (plastic bowls especially).
Now you could buy a new non-plastic bowl, buy a bowl or fountain that recycles and filters the water, clean the bowl and replace the water more often, or simply let your cat continue drinking from the faucet. Some people just leave the faucet on a light stream for their cat but that can waste a lot of water over time.
Instead of leaving your faucet continually running, you can buy an automatic sensor that attaches to your bathroom or kitchen faucet. These devices may be advertised for use by us humans but can also be used by your cat as seen in the video above.
The adapter above simply attaches to any standard faucet as would a water filter and is powered by batteries. You adjust the flow of water that you want. When it senses your cat near it, the water starts flowing just how you set it.
Because this adapter starts the water flow when your cat is near and automatically shuts off the water when your cat leaves, it can preempt or compensate for a cat who has already developed a habit of turning the water on using the handles.
The AquaPurr is another option for drinking water directly from the faucet. This product doesn't require your cat to go in the sink. Instead it creates a platform on the side of the sink where the weight of your cat on that platform turns on the fountain.
It requires no batteries but does require some space to the side of the sink for the platform to sit. Hopefully your cat won't get comfortable on the platform and take a nap there (which would keep the water running).
A newer version of the AquaPurr (AquaPurr eC) removes the platform and the space requirement for it by using a sensor to start the water flow when your cat is near it.
If you do encourage your cat to drink from the faucet you should make sure a backup source of drinking water is available (e.g., water bowl) in case the faucet becomes temporarily unavailable for some reason (e.g., batteries die, power goes out, water gets shut off, bathroom door is closed, etc.).
The product solutions on this page all require connection to a faucet. This means the products are expecting a "standard" connection.
It's doubtful you'll find this connection in a tub so we're probably talking about a kitchen or bathroom sink (although you may also have a utility sink located elsewhere that may work).
Below is a video of how the iTouchless faucet adapter is installed. Notice the removal of the aerator before the adapter is installed. The aerator is simply unscrewed from the faucet and the adapter is screwed in its place. It's this connection that needs to be standard. If it's not then you'll need a connection adapter (which may come with the product or you can check the home improvement store). You could also just use a different sink that has the appropriate connection.
If you have the ambition, you could also replace the whole faucet with an automatic one. Your new faucet would then have built into it the functionality provided by the automatic water shut-off adapter.