Three-way circuit where one switch also toggles an extra light fixture on/off

Question

I'm trying to figure out a circuit in my home that operates two sets of lights: a set of track lights over the stairwell, and a light fixture at the end of the upstairs balcony. At the bottom of the stairs is a 3-way switch, at the top of the stairs is a 4-way switch, and at the end of the balcony there is what appears to be a 3-way switch. When the switch at the bottom of the stairs or the top of the stairs is flipped, it turns the stairway lights on/off like a regular 3-way circuit. The switch at the end of the balcony also turns the stairway lights on/off like a regular 3-way circuit, but at the same time it turns the balcony light fixture on/off. By flipping the balcony switch you might be turning both lights on, turning both lights off, or turning one on and one off. The two stairway switches have no effect on the balcony lights.

I have two questions about this setup:

1. How is the circuit wired to produce this behavior from the balcony switch? I do understand how a basic 3-way circuit works, but I don't understand this extra wrinkle.
2. Would it be easy to rewire the circuit so both light fixtures turn on or off together and are controlled by any of the three switches? Or failing that, to rewire the circuit so the stairway lights are controlled by the switches at the bottom and top of the stairs, and the balcony lights are controlled by the switch on the balcony? Easy is the keyword here; I'm not able to run any new wiring, only to change connections inside the junction boxes.

EDIT: As requested, here's a truth table for the possible switch combinations. S1 is a 3-way at the bottom of the stairs, S2 is a 4-way at the top of the stairs, and S3 is a 3-way at the end of the balcony. L1 is the lights over the stairway, and L2 is the light fixture at the end of the balcony.

S1     S2     S3     L1    L2
down   down   down   on    on
down   down   up     off   off
down   up     down   off   on
down   up     up     on    off
up     down   down   off   on
up     down   up     on    off
up     up     down   on    on
up     up     up     off   off

S1, S2, and S3 in combination work like a normal 3-way circuit for L1. Regardless of the positions of S1 and S2, S3 works as an on/off for L2 with down=on, up=off.

EDIT 2: I added photos of the switches below. For each photo, I circled which wires are grouped together in the conduits. The only difference I see from a standard 3-way circuit is that Switch 3 has an extra black wire connected to a traveler screw.

Switch 1:

Switch 2:

Switch 3:

Answer 1

From the truth table it appears that S1, S2, S3 together are a correctly-wired 4-way for controlling L1 over the stairs. The extra wrinkle with L2 on the balcony implies that power is supplied into the circuit at S3 (the balcony end of the chain) and switched-power out to the L1 stairs light is at S1. You can confirm this with a volt meter or maybe an NCVT: check that the common terminal at S1 is live always and only when the L1 is lit, or that the common terminal at S3 is live always regardless of the positions of any switches.

The S3 switch on the balcony is being used as a 3-way switch for the stairs light, but at the same time it is being used as a single-pole switch for the balcony light. That extra black wire off one of the traveler screws feeds power to the balcony light according to the position of S3, without any relation to what's done with S1 or S2.

As for what other behavior you can get from this circuit -- that depends on how the switched power at S1 makes its way to L1. Given there are two black wires and one red attached to S1, I infer that cable or conduit goes from S1 directly to L1. (In other words, the switched-hot does not return to the S2 or S3 locations.) There's probably no way, without adding wire, to make L1 and L2 be switched together.

You could easily accomplish the second-best outcome, which is that L1 over the stairs is controlled by S1 and S2 while L2 on the balcony is controlled by S3. Pick one of the travelers at S3 and move it to the common terminal. Disconnect the other traveler and cap it (with a wire nut for example). Done. Leave some kind of indication for the next person so they can figure out that the red and black pair are supposed to be travelers..

An alternative, and arguably better solution, is to bring in a "two 3-way switch" device. One example, Leviton 5243, is pictured below (photo: leviton.com).

In this way you can retain S3 which continues working with S1 and S2 to control the stairs light, while also fitting a new S4 for controlling only the balcony L2.