I've seen this joint called a "tusk tenon" or a "keyed mortise and tenon".
The "tusk" is the vertical wedge.
The secondary mortise that the vertical wedge goes into is the "key mortise".
So I guess that means the tusk could also be called a key.
A "wedged mortise and tenon" is a different joint, it involves cutting a mortise and tenon so that the mortise is tapered, so the tenon slides easily into the mortise, but with slits cut into the tenon that you then hammer wedges into to expand the tenon while it's inside the mortise and lock it in. This is generally a permanent joint, unlike the tusk tenon, which can be knocked out for dissassembly:
All that said, language is seldom ever as clean, simple and organized as people would like it to be, so it's quite possible someone, somewhere in history called the one joint by the other's name, and you can find plenty of people calling the tusk tenon a wedged tenon.
I've seen a number of ways to cut mortises.
The main one people recommend is drill out the bulk and then chisel out the rest. It's annoyingly hard to do a clean job in pine, I guess the grain is too large and wavy because it tends to split. I've done that I guess 20-30 times and it's no fun so I've done a lot of looking into alternatives :-).
Other options are:
A router and a template and clamps, maybe with a chisel to clean up the corners later.
Note, a neat trick for routing deep mortises is to route from both sides with bits long enough to overlap, doing the first side a top-bearing bit that rides along the template edge, and the other side with a bottom-bearing bit that rides along the clean bottom edge left by the top-bearing bit.
A slot mortiser, which is basically a router in a specialized jig to make it easier and faster.
A hollow chisel mortiser, which is basically a drill press with a drill bit encased in a four-sided square chisel. In the videos these all work like voodoo but I'm always skeptical of demos. I've never used one, but as far as I can tell this is just a glorified version of drill & chisel by hand. I.e. the chiseling aspect is still pretty much the same, though the lever action and the drill press mechanism maybe make it easier.
Note, there are also slot mortiser add-ons for regular drill presses, but every comment or review I've ever seen of them has been negative.
A chain mortiser, which is basically a plunge chainsaw. Very popular in Japan.
Or if you want to make large mortises and have more money than sense and 3-phase power a swing chisel mortiser (which is how door factories mortise out the doorknob mortises).
Or of course a CNC router. With a CNC you could use a smaller bit and get sharper corners. Using a router normally the rule of thumb seems to be that you need a router bit that's the width you want your mortise to be, which of course means the larger your mortise width, the larger the rounding is in the corners.
The traditional way, using a straight-sided mortising chisel, check out this very cool video by Frank Klausz:
I have a nice Sorby mortising chisel and I've experimented a little with it but it's slow going in harder woods. That may because they're hard :-). Also, I'm generally doing very large mortises compared to most people (1" x 4" x 4", etc). Or I may just need to work more on my chisel sharpening skills.