Bruce, the flat thin stuff is very often called bezel strip. I get the silver version from Hoover & Strong in NY. They may have it in gold on their website. I use 30 guage for inlay.
You don't want to cut the groove in. You want to chisel it and not remove any material. All you're doing is separating the fibers so they will swell back up around the strip when wetted and hold it in place. Don't tap the strip in too hard or it folds over and leaves thick wavy sections in the lines, becoming a royal PITA. If your wood is good and dry when you start, you'll be surprised how much it'll swell back around the strip. When done right it's hard to pull out prior to sanding flush. Like said, the chisels are often made from little pieces of re-ground hacksaw blade; my favorites are 1/8 and 1/16" wide at the tip. In his video, Alan Aldridge mixes a drop or two of woodworker's glue on top of the wetted wood and smears it around on maple. On woods that won't swell like maple, use superglue.
BTW, order it hard or half-hard. I don't personally like soft wire, it wavers too easy. For tight curves, where you need it soft, anneal it with localized heat from a lighter or small butane torch.
