As you can see from the WR dictionary definitions, "solder" isn't used as an occupation name, so you can't say someone is a "solder" in the same way as you could say he/she is a "welder".
Soldering and welding two items together are quite separate techniques. To solder two things together is to join or fuse them using an alloy solder. To weld two things together is to soften the two ends by heating and then fuse or hammer them together.
I think where you were getting confused was with the base verbs. You don't "sold" things together, so someone who does that wouldn't be a "solder". You "solder" things together, so it has to be a "solderer".