On your new switch, the 2-Way is for when you have two switches controlling the same light such as on a stairway. Just ignore those in this instance.
The live wire in goes to the connection marked COM. The short length of wire that fell is to join one COM to the second COM.
The feed to one light goes into the 1WAY terminal on one switch and the other goes to the 1WAY terminal on the other switch.
The same applies to the ols switch that you have put back in, but with 3 terminals top and bottom, it looks as if ine if the switches is inverted so one has COM at top and 1WAY and 2WAY at the bottom. Beside it the other switch has 1WAY and 2WAY at the top and COM at the bottom. The same wiring connections apply.
WARNING. Don't forget to isolate the circuit before working on it. Switch off mains / remove fuses or circuit breakers. Test with a meter to be sure.