Your Plan
Your analysis about two wires makes an assumption that is forbidden in residential electrical work. One of the wires in a standard fixture is the neutral whose state cannot change. It must always be connected to the other neutrals in the system, connected to the main electrical panel and ultimately to ground. If 1 reresents connected, the white wire is always in the 1 state.
The black, or hot wire can be connected or not. Similarly, in a three wire system, the red wire, as well as the balck, can also be used as a hot wire and can be connected or not, depending on the state of switches. In a three wire system, the white still stays connected all the time.
How to do it
If you want separate controls, you need a separate wire to power (carry the hot current) to each bulb. A single neutral (white) can be used.
Hot wire originating at switch
If the power is presently comming into the switch box from the electrical panel, and then being routed to the fixture, the standard way to wire this would be to run a three wire cable (white/black/red) from the switch to the fixture and to use a double switch, such as this type.
The white wire coming from the panel is connected to the white wire going to the fixture. The black wire coming from the panel is connected to the incomming sides of both sections of the switch. The black wire to the fixture is connected to one of the outgoing terminals and the red wire from the fixture is connected to the other outgoing terminal (outgoing because the power is coming through the switch and going out to the fixture).
Hot wire originating at fixture
If the power comes directly to the fixture, and there is just a two wire cable going to the switch, you need to run a three wire cable to the switch. The white wire will actually be used as a black/hot wire to carry power to the switch. It needs to be marked with black tape a both ends to show this. At the fixture this pseudo black wire is connected to the black hot from the main panel. The real black is connected to one bulb and the red to another bulb.
At the switch, the white (now marked black) is the incoming line to both switches and the red and black are attached separately to the outgoing terminal on each switch.
In either setup, the switches are independent. You can turn on either, both or neither.
There are double throw switches that would allow you to have one or the other or neither, but not both. You still need the three wire cable at the switch site.
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.
Best Answer
If those three wires are the only ones in the junction box, then you're out of luck. The red wire and the black wire are travelers going to the other switch. The white wire appears to be connected to the black, common, screw so it's going to be a switched hot going to your light at the bottom of the stairs. There is no always hot or a neutral so adding a new light is a no go. Can you run wire from your existing light to the additional location?