There are several different Rodeo models - which one is yours? The wiring is different for a hard top vs a soft top, the S model has different wiring, etc. So we need to establish that first.

I do know this - the high mount stop lamp and the lower stop lamps share the same ground point, so it isn't likely to be a loose ground, else the high mount stop light would be affected too.

I also know that the turn signal switch WON'T be the problem because the turn signal switch on that vehicle has nothing to do with the stop lights - it's a separate circuit.

What this does sound like is a problem with the bulb sockets or the bulbs themselves - I'd start by replacing both rear bulbs, because that's cheap, and a ground problem can happen right there at the bulb. And those bulbs can look good and be bad (the 3357 and similar varieties). Examine the bulb sockets for corrosion or heat damage. It would seem odd that both sides would have the same problem, but they could have died one at the time.

The reason that the cruise control would react to this concern is that the cruise control searches for a ground through the stop lamp bulbs and when that ground is neutralized for some reason, (bad bulbs, bad bulb connection, bad bulb ground, etc.) the cruise will react as if the brake pedal is being activated, which will kill the cruise as well.

The brake lights may be getting their ground through the tail lamp filament (or vice versa), which would explain why those lights go off when you apply the brakes.

Let's replace the bulbs and examine the sockets first. That'll be the cheapest thing to do. Let me know what happens...