Electrical – 3-way switch with 2 lights, alternate off/on

First, just to prevent any confusion, it's a "3-way" switch setup, not 2-way. 3-way switches take power in (or let it out) through a singular wire, which then is switched between two 'traveler' wires. It's referred to as a 3-way switch for the same reason as a 3-way intersection - there are 3 paths in/out of the switch.

Next, you need to ensure that the wires leading to and from your switches actually go to the light. This probably goes without saying, but if you've changed the light's wiring as well then that could be the issue.

However, the final solution will always come down to the wiring or the devices. The light itself does not control switching, so it is not bad. You might have a wrong switch, bad switch, or again just have some wiring incorrect. Follow this diagram below and it'll work:

The diagram might not be the exact layout for your situation as the power could be coming in at the light or the light could be between the switches. Regardless, that would simply call for wire nut connections and it would still come out to this exact setup.

Go for the Pro model

I spoke with Lutron's tech support online via livechat and the Lutron Pico remotes do support being paired with more than one Maestro or Caseta dimmer. With the correct parts, you also can leave the mechanical switch in place upstairs to allow a houseguest or first responder quick control of the lights.

You'll need a black pigtail (match the existing wire gauge) and a few more wirenuts for this, as well as two Lutron PD-10NXD (this is the pro model, which adds the hardwired switch support) dimmers and a Pico remote with a wall mounting kit.

1. Turn off power at the breaker (if it isn't already off that is)

In the upstairs box:

2. Remove the two black wires from the old switch and nut them together with the black pigtail.
3. Attach the other end of the black pigtail to the black screw on the upstairs switch. (This effectively uses the 3-way switch as a single pole switch.)

In the downstairs box:

4. Remove the top (in your photo) black wire from the left 3-way switch. Nut it to the red wire on the dimmer that's going in the left side.
5. Remove the top (in your photo) black wire from the right 3-way switch. Nut it to the red wire on the dimmer that's going in the right side.
6. Pull out the neutral bundle from the bottom box. Connect the white wires from both dimmers to it. (This will yield improved dimming performance and avoid flickering or other misbehavior if CFLs or LEDs are installed.)
7. Remove the bottom (in your photo) black wire from the right 3-way switch. Nut it to the black wires from both dimmers.
8. Remove the red wire from the right 3-way switch. Nut it to the blue wires from both dimmers.
9. Pull out the ground bundle (if present) and nut the green wires from both dimmers to it. If no grounding means is present in the box, nut the green wires to each other but do not connect them to any other wires in the box.

Back at the upstairs box:

10. Install the Pico remote and wall mounting means adjacent to the upstairs switch box.

11. Button both boxes up.

12. Turn the power back on at the breaker.
13. Pair the Pico remote with both Caseta dimmers as per the supplied instructions.