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I am trying to come up with a joint that I can use for a frame that will have beveled inside corners, where I can cut the bevels cleanly on my router, and that isn't a miter (I don't want the 45 degree seam, I want a 90 degree seam).

The only joint I can really think of is this (inside bevel on horizontal piece not shown because it's not really significant to the joint):

enter image description here

My question is: Does this joint have a name? I know I've seen it before, but I couldn't find anything in image searches, and unless I overlooked it I don't see it on this huge list of joints, either.

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It is a coped joint. You make it by cutting the adjoining piece at a 45 degree angle, then using the cut line as guide to remove the area below the line.

Coped Molding

Here is an image of how to do it.

enter image description here

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    If you've ever wondered why it's called a coping saw, this is it. [Some day I'm going to build a display frame containing a saw and the single word "Cope." With coped trim, of course.] – keshlam Dec 1 '15 at 3:19
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    This is not a coped joint. A coped joint meets the adjacent surface at a sharp edge as your second illustration states. That won't be the case here. – Graphus Dec 1 '15 at 10:12
  • @graphus I am confused now. I am joining the two pieces at 90 degrees, and I actually had moderate success creating the joint as in the picture (although it felt like too much effort for my simple profile). But the beveled edge marked with text in my picture actually was significant, at least for this method of cutting the profile. The picture here does seem to match what I'm doing though, I think. – Jason C Dec 1 '15 at 23:18
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    @JasonC, instead of us focusing on coped/not coped I think you should just do this differently. So do the corner joints as you like (e.g. dowel- or biscuit-reinforced butt joint) and then form the chamfer on all the inside edges in a single operation. The corners will need finishing off with a sharp chisel of course but this is the way this kind of thing is nearly always done, example, – Graphus Dec 3 '15 at 11:39
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    @Graphus Ah. That makes sense. That's actually what I was specifically trying to avoid, I wanted to make the inside bevels first along the full length of each side and avoid chiseling out the corners. I was looking for a joint I could cut quickly without putting much thought into the bevels. But its starting to look like bevel then cleanup is the easier way. – Jason C Dec 3 '15 at 21:16
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Making frames like this is typically done with a "Stile and Rail" router bit set. The bits come in many different configurations and can vary wildly in price, but this is how door and window frames are made. The additional tongue and groove provide a lot of extra strength that your diagram doesn't show.

They come in many different profiles, and the image below is just a random sampling sold by http://www.mlcswoodworking.com (no implied endorsement - never used them).

As Graphus pointed out in a comment, this really isn't a coped joint. I'm not sure that it has a name other than "stile and rail joint".

enter image description here

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Is the "rail and stile" look what is important or could you bevel both pieces first and miter the joint? I just did some cabinet doors where I set my table saw to 45 degrees and cut, what is your left piece, to the depth of the bevel, a miter). Then ripped the end of that piece to that point. On what would be your right piece, it would be beveled first, and then miter the tip of the bevel to where the end of the bevel will meet the left piece. The bevel is mitered joint, and the rest of the joint is a butt joint. I then doweled the joint so there would be no visible fasteners.

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