Grats on realizing the importance of the screw colors. That matters. Position doesn't.
Now, the problem is, it's clear that travelers are not properly identified. Now, I'll tell you a Harperism: I like to positively identify travelers, and when I know what they are, I mark them with yellow tape. Both travelers can be yellow, they are interchangeable and don't need to be distinguished from each other.
The problem is, your travelers are a big mess right now, and they don't agree. On a 3-way, both travelers go on the brass/gold screws (hence my yellow). On a 4-way, one pair of travelers goes on brass, the other pair goes on black.
So the travelers are red+black on the 4-way, red+white on one 3-way, and white+black on the other 3-way???? They are different colors on every switch. That's just crazy.
So... how do we tell which ones are really travelers?
- Each cable has two ends. The travelers are the same colors on both ends. If they're red and blue at one end, they'll be red and blue at the other end. (mind you, we don't know how the wires are routed here).
- We know that both travelers are ALWAYS in the same cable. One 3-way or the other, will be fed by 2 cables. The 2 wires in the same cable must be the travelers.
- If both 3-ways are fed by a single cable, then go up to the lamp. There'll be two /3 cables, with 2 wires from one wired to 2 wires from the other. Those are the travelers.
- In rare cases that funny splice can happen in a junction box elsewhere.
Once you have identified the travelers where it's possible to identify them, then you need to move down to the 4-way. Figure out which of the 2 cables inside the 4-way connects to the
Hot or not
The thing about travelers is... one is hot, and the other is not. And that's supposed to swap when you throw a switch.
If you see a wire in one of the boxes that is hot at all times, that is the supply always-hot. There must be one of those in the system. Obviously, that's not a traveler.
If you open up the boxes and pull the switches out without unhooking anything, and got us some well-lit photos of what is connected to the switches and in the back of the box (diffuse daylight is much better; flashes cast shadows that make it hard to see wires)... then we can give more specific advice.