3-way switches, one with an extra (non-ground) wire

I'm putting new switches in my new (old) house and I continue to have bizarre situations.

One light is connected to two 3-way switches. Switch 1 is pretty straightforward: black hot wire, white and red travelers and green ground. (Wires laid out without switch in pic 1088 below) I used my tester to see that the black wire was the hot wire, and swapped that switch accordingly. Testing it before I replaced switch 2, everything worked fine (see pic 1090 and 1091). Switch 2 is different: it has two white and two red wires (see pic 1089). The old switch had four terminals, which I found odd. When I touched it initially with my tester, the two white wires were both hot. So I pigtailed them together and connected them to the hot terminal lead . Then the two red wires I connected to the brass terminal screws. (in pics 1093 and 1094) The weird part was that after doing so, Switch 1 worked just fine. But without Switch 1 being in an 'on' position, Switch 2 would not work. That seemed like an easy fix, to swap the traveller wires on each side of Switch 1 (moving the red and white wires opposite each other) but doing so did nothing to change the situation where Switch 1 had to be in one position for Switch 2 to operate (see pics 1095 and 1096). I then pulled the wires off of Switch 2 and tried again with my tester. This time, one of the red wires tested hot. So I moved the red wire to the black terminal screw, put the travelers on the other terminal screws. No change. And I then swapped the two traveler wires from one terminal to the other. No change. With four wires, I tried a 4-way switch. No change. How can I rewire this to not make Switch 2 dependent on Switch 1's position?

 

Do you have a picture of the old switch? Four terminals means it was a four way switch. Is there a third switch that control this light?

 

Definitely a 4-way switch. So I imagine I should swap in a 4-way?

 

Yes replace four way with four way.