What you’re describing sounds really typical for a 1960s or 1970 US double switch without a neutral.
Remember that in the United States, wire colors are not mandated code. People can and do use whatever bits of wire they have lying around.
For a while, you used to see red as the hot wire coming into the switchbox, black as the load going from one of the two switches to one device, and white as the load going from the other switch to its device.
Note that this is not a three-way switch. It’s not that both switches control the same device. Rather it said there are two devices on the same circuit, each with its own switch. Really commonly used for, say, a bathroom light and fan. In the garage you would sometimes see this set up for interior lights on one switch and exterior lights on the other. You also see it in utility closet that had one switch for an overhead light and another switch for some device in the closet, typically a water heater timer. You also sometimes saw this for pools, one switch for the lights and one for the pump.
I want to emphasize again that there is no neutral at the switchbox in these configurations. But there has to be a neutral someplace to return the current to the circuit box, and that most typically was done at the light fixture and the other device that is sharing the same circuit. Doesn’t really matter where it is, it just has to be somewhere.
Here is one diagram of this model. It’s a dual switch in a utility closet or basement, one switch controls a fan, the other switch controls the light. No neutral at the switchbox: the neutral is at the light fixture.
I can’t say for sure that this is what you’re seeing, but it sounds like it.
Red is the hot coming into both switches. Black is the load out from one switch and white is the load out from the other switch. However some people would use black for both of these. And starting around 1980 the colors were usually done in reverse, with black used for the hot and red used for the load to the light fixture, but again there’s no code requirement and people can do it however they want. And just for fun, but again really really typical, white is also used for the neutral that connects at the light fixture. So to “avoid confusion” The white line in from the switch to the light fixture gets taped black where it comes into the light, so now the light fixture has a black wire in which is the same as the white wire coming out from the light switch. And the light fixture has a different white wire going out.
(All because electrician kits used to just have White, red, and black wire. In the 21st-century if we were designing national code to address wire colors we’d probably have eight different colors and use each one consistently. But in the United States we do not have required wire colors. So always tread carefully.)
Anyway just one possibility. You need to test all the lines to be sure. I know you said the second switch was a different circuit, which would make the wiring a little different, but the same model might apply as far as red in, black out, neutral at the fixture. (Having two different circuits in the same switchbox adds a whole bunch of wiring issues, and you don’t usually see it these days, but people did used to do it so you can run into it occasionally.)
Also, I know you’ve already looked at this but just for completeness while we’re adding other links, we should include the link to the wiring FAQ: