Edge rusher is almost a seperate catagory from DE.
Edit if you are including LBs then I'd move out Beasely and include Terrell Suggs in the final 4
The term EDGE is a recent one that includes both rush ends and outside LBs who tend to rush the QB. Sure, a strong-side DE is going to be tasked with holding up strong against the run, dealing with a tackle and often a TE to hold the edge. But the backside DE is usually an ears-pinned-back rusher getting around the tackle, most often on the QB's blind side.
Here, let me make a visual really quickly:

Edges are the guys tasked with pressuring the QB (in red).
Your strong-side end in the 4-3 is your 270 lb guy who can man up against that run blocking. Your back-side end is your 245 lb guy coming like a bat out of hell, chasing down the QB.
In the 3-4, your ends are 280 lb and more like a 4-3 DT, all 3 linemen holding up against the run, since there's only 3 of them. They're freeing up the LBs to either make tackles or, like the outside LBs, get to the QB.
I made the Will (weakside LB) in the 4-3 purple, because unless they're getting a slot WR to cover inside the hashmarks, they're often freed up to rush the passer and tend to get a lot of sacks, almost by chance. If that DT in front of him is good, there's just a red carpet rolled out there, splitting the G and T for him to get up through there.
And I could have made the strong-side DE purple in the 4-3, if the offense isn't using a TE much or the TE is a real pass-catching threat and not much of a blocker.
In football video games, they'll label a recruit or draftee either balanced, run-stuffer, or pass-rusher, based on his size and skill set. EDGE is the pass-rusher.