The bent frame pictured is not necessarily "double butted".

What you have there is a "lugged" frame (bent badly, of course). The tubing is brazed (like soldering, but with a brass alloy instead of tin/lead solder) into the lugs that form the frame.

Butted tubing refers to the inside diameter of the tubing itself. "Double butted" tubing will be thicker at the ends of the tubes, where the meet up with the lugs (for example) and thinner in the center section. This allows for very light weight, yet still strong.

But to answer your question, "yes", you will want a double butted frame. You'll probably want it to be a lugged frame also, as they are prettier than non-lugged (in MY opinion... someone will undoubtedly chime in to the contrary)

EDIT #1: Whatever frame you find, DON'T cut anything off! At least allow it to be converted back to a multi-geared bike.

EDIT #2: Try to find one that is your size, and make it look like a proper bike. Too often, I see guys running around on frames that are WAY too big for them, and they've got the saddle slammed all the way down to the top tube.