Have a look on this site at a diagram of your headlight assembly and hopefully you can work out what part has been cut out.
Yamaha YBR125ED 2009 3D99 EUROPE 1H3D9-300E1 34 HEADLIGHT - buy original 34 HEADLIGHT spares online (cmsnl.com)
I would of thought the bulb is a two filiment type with a dip and high beam so at least two wires possibly three if the bulb doesnt earth through the headlight mounting. Have a search for a wiring diagram for your model so you know what you are looking for. Good luck.
If it cut out when it started to rain then water is getting at th ignition somewhere so try to dry all the connection blocks out check the wiring for damage or water ingress. Look at the HT lead and cap and dry out and seal lead and possibly change plug cap if it looks suspect. Dont understand why there was water in the airbox but check dry out if water in there and dry filter or replace. It may pay to drain all the petrol out of the tank as there could be water in there and carbs injection systems do not like droplets of water. Once you get the ignition dried out get a spray can of ignition sealer and spray on exposed leads and tape if anything looks bodged. If ts got a injection system it probably doesnt have a choke so the ecu will adjust mixture from cold. Look up the cold start procedure some bikes need the throttle closed others you have to open slightly, what does the manual say on starting.
