What makes @NomadTech assumptions on your existing wire strange (but possibly correct) to me is why would the electrician wire it that way when the Live Hot is sitting right there in the box?
Just for my clarity, are the pictures before you did any attempts correct? And if yes then these pictures are of an actual working 3-way light setup BEFORE you attempted to make changes so you are wanting to replace the switch you labeled “Lights” in the 3-gang box with the primary and the lone single switch in the photos to be the add-on? If yes then we are trying to verify what you probably have is a variation of line-switch-load-switch-2 example in the GE 3-Way FAQ with that red wire of yours being the primary hot. So IF when you meter out that red wire and confirm that is a direct feed from primary breaker power you can wire it like shown in the FAQ but where the image’s black line power is your red.
