On my bike I have a bottle dynamo rated at 6V 3W. It has two connectors, + and ground. My front light is an AXA lamp that I'm not sure of the model, but is very similar to an AXA Pico 30. It has a switch on the back (on/off, if I remember correctly from the instructions it refers to whether the second connection acts as ground or input from a rear light). It is rated at 6V 2,4W. My rear light is a very cheap model that mounts on the luggage rack. No ratings written on it. Both of them use an LED. When I first installed them I connected the + of both lights to the + on the dynamo and ran the grounding wires to the bike frame. They both worked together. The problem appeared last month, when the rear light stopped working. Only the front one lights up and sometimes, very briefly, the front one turns off and the rear one lights up. Right now the wire configuration is in parallel, + to + on dynamo and ground to ground on the dynamo for both. Still the same problem. When only one is connected they both work fine, so the cables are alright. I tried unmounting the rear light from the rack (thinking the screws might short it to ground), connecting them in series and using that switch on the front light, inverting the terminals in both series and parallel configurations. The rear light simply refuses to work. I measured the output of the dynamo with a multimeter, when pedaling really fast it gives around 15V and 400mA AC. Connecting them one by one, I calculated that the front light draws 2,4W and the rear one around 1W. What do you think it might be the problem. I'm really at a loss right now.
The front light looks like its rated for Dynamo use, so will have internal protection from over voltage and AC input (if it is expecting DC). The rear light (cheap/unbranded) possibly does not and will likely be the cause of the issues. Even though it lights up, does not mean its not got something blown up. Could be as simple as its blown a component in the Buck/boost circuitry and is now less efficient, drawing more current and that's enough to stop the front light working reliably.
Dynamos on bicycles are pretty crude devices, and LED light a pretty advanced device. Mixing them unless the LED is rated for dynamo use is not likely to be successful. Personally, I would stick to the same brand Dynamo and lights if I were to go down that path, unless the light manufacturer explicitly specified 'any dynamo.'
I would replace the rear light with a branded one specified as suitable for dynamos.
If your front light has connections for a rear light, it's wise to use them. My Axa Luxx 70 does. So does my cheap nasty Raleigh front LED lamp. Note that in this case the polarity of the connection to the rear might matter; otherwise it doesn't as dynamos are actually alternators and put out AC, thus the lights have to rectify this (or only use half the waveform, relying on the LED itself to be the rectifier.
I've these front lights with a B&M Secula and a couple of cheap unbranded rear lights, always connecting through the front light.
It's always worth running ground wires if you can - if you only have 2 single-wire terminals on the front light, they're V and GND. Connections through the frame are inherently unreliable.
Connections are in parallel if not routed through the front light. Connecting in series you're likely to get nothing at low speed, and at high speed either a very dim front or (less likely) a bright and short-lived back. The back needs around 1/10 the current of the front, and a series connection means they have the same current going through both.
Are you sure you didn't press the button accidentaly?
I had a similar Setup, and the button on the AXA was working as a switch for the rear light only. Sometime someone pressed it and the rear light stopped working.
I Was about to rewire everything from scratch before I decided to test it. It seems however you are already past that point...