So, watching the 2nd video, there doesn't seem to be any rack movement as it slows up, so I think we can rule out a problem with the governor. I very much doubt the air filter is the problem, as it looks to be an oil bath air filter - if that were acting up I'd expect it to over-speed on ingested oil. Which just leaves the fuel system....
Listening to your first video, the combustion sounds kind of weak, and after a few pops, it seems to stop entirely, before trying to strike up again. So for me, that's definitely a fueling problem.
First thing I'd do is check out the filter. You say it hasn't run veg for a while; but unless you've thoroughly cleaned the lines, there might be a gob of the stuff blocking up the injector. I'd be tempted to start by disconnecting the injector fuel line: Crank her over a few times & make sure you're getting a dollop of fuel every other revolution. If that's happening, then you're definitely in injector territory: Pull the injector, re-connect the fuel line - then (with the injector safely pointed into a catch pot of some kind, and definitely nowhere near your skin!), crank her over. You should get a nice fine spray of fuel every other rotation. If it's looking a bit dribbly, or inconsistent in volume, or it's not injecting every time - then that's your problem...
If, on the other hand, you're not getting your dollop of fuel out of the disconnected feed line; then either there's air in it (keep cranking & see if it steadies up), or there's a problem with (most likely) the fuel pump: Either it's worn (and letting fuel past the piston instead of out into the high pressure line), or there's something preventing enough fuel getting into it. If it ran OK on veg & not so much on diesel, it COULD be wear, since veg is thicker than diesel. Another possibility is it's not shimmed correctly (so is under-fuelling), or the cam lobe is worn (= under fuelling). Both could be fixed with more shims, although if it's a worn cam lobe, you're only putting off the inevitable replacement...
However... my money's on the injector.