Except when I run water in the bathroom sink; she'll drink that in front of me. But you don't have a "strange and mysterious cat." You have a cat. They are all weird.

My cats don't have a problem drinking in view of humans but they quite often disregard their designated water source for another outside of my view. I also have one who I refer to as "soggy paw". More often than not he will drink by dipping his paw and licking it. It's always the right paw so I guess he's not ambidextrous.

We have 2 new kitties and one of them only drinks in front of me by lying next to the bowl, dipping a paw and licking it. Dips his paw into my water glass if he can get close to it.
Only cat I have ever had who does that.

Haven't tried running water, but he seems very fascinated as I pour water into his bowl.

2 male neutered cats, indoors only,, and I have, from experience, a concern about urinary problems, so try to keep track of their water use.

I had a Siamese that would only drink from the aquarium, and mostly from the area where the aerator pump was bubbling. Maybe it's taste, maybe it's an instinct that it's healthier to drink running water. Another cat wouldn't drink from a bowl unless she heard/saw the water being poured into it, and would jump up on the vanity beside the bathroom sink if she heard the water running, and try to get a drink there. I regularly had to let the water run in the sink with the stopper in so she could get a drink from it.

He would dip his paw in the water and then lick it. He lived to be 15 and he always drank water that way.

They love it. We see them drinking all the time now.

That can sometimes be a turn-off to cats who don't want to drink where they defecate.

We have 3 cats, they are older. Two are 17. They love that bubbler.

He and his brother were adopted from a home that only fed them Purina Kitten Chow. I switched them to Purina One Kitten and thought they'd love wet food since they never had it. Nope. Tried all different kinds and they'd take a lick or two and then gobble down the kibble, which I was free-feeding (mistake). Switched to Indoor Advantage at one year.
At their checkup Charlie weighed in at 16 lb. and Andy not much less. Vet said to limit them to no more than 1 cup dry per day per cat and then they'd be hungry enough to eat some wet food. As soon as I started the "diet" Charlie started knocking their self-watering device over and I'd come home to a wet kitchen floor. Back to drinking out of a cereal bowl. Finally found a canned food they would eat (Fancy Feast in the blue cans) or at least Andy would eat. Charlie would sniff it and walk away. But I have caught him eating it when I'm not in the kitchen. I think he will not eat it in front of me because he doesn't want me to think I "won" the food rebellion.

Our kitties had to share their water bowl with a dog for most of their lives. And we'd have to separate
their food dishes so the dog wouldn't eat the cat food, although Mouse (the kitty) used to like to eat the dog food.

Now we are down to just a dog. She has no problem eating or drinking in front of us.

and he doesn't want to tip anyone off as to its whereabouts..

THIS is strange:

Whenever I went to the bathroom, she would dash in the room and jump on the sink. I would run the tap a little and she would put her head under and drink the water. She would drink out of the water dish as well.

...drinking from any sink faucet. They do not seems to mind if their heads get a little wet. The girl will actually come into the large walk-in shower, sit for a while at the far end, and then come closer to the water. She doesn't mind having the hand-held spray swooshed across her back sometimes.

Another one has figured out how to pull down the handle on the Hinckley Springs water dispenser, and will get himself a drink whenever he feels like it. he also comes into the shower when I am done, and laps water off the shower floor.

The 4th one drinks from the water bowl, but in a strange way. She is our smallest cat - not quite 6 pounds all grown up. She stretches her little neck all the way across the water bowl to drink from the far edge. No idea why

I can't leave the toilet seat up as the cats want to drink out of it. They even try when I'm using the toilet. In their defense, they were abandoned and survived by drinking out of the toilet, but it's still a little weird.

It took me months to get one of them trained to tell the difference between their litter box and my bathtub.

Fortunately they make up for their idiosyncrasies by being utterly adorable.