You missed one important part of that video...
A recording is a sum of its parts (not just the mic used) or, more accurately, only as good as its weakest part. That isn't to say the mic isn't important, but equally important is the instrument, player, playing technique, recording technique, recording apparatus, mixing and any processing. I should think the amp used for that video was a very weak link in the chain, and also the recording sounded like it could have benefited from some compression to smooth out the levels (or some more consistent playing to begin with).
If you like the sound of your current amp setup, throw a mic in front of it and chances are you will like the sound of it recorded... it may require a bit of experimentation with mic placement, etc, but if you already have the tone and overall "sound" you like then its just a matter of working out how to best capture that.
Alternatively if you're open for something a bit different (and possibly more versatile) then DI recording might be a better direction. Amp modelling has come a long way and you can quickly dial in some very passable tones with lots of flexibility, or you can shape the DI'd signal using just EQ and compression. Some people don't like DI recording, but personally its my go-to for bass (not so much for guitars).
Then you can always do both and blend to taste, but that's straying from the point a bit
I think the best starting point would be to chose a small 2-channel USB or Firewire interface (with mic preamps) and then decide where you want to go from there. If you want to go the DI route then many of these interfaces have high-impedance "instrument" inputs built into them so you don't need a DI box, but if you decide to record with a mic then you can just plug it straight in... and obviously this gives you the flexibility if you want to change your mind later / try both.
You can pick up little interfaces like the Presonus Firebox or M-Audio Fast Track Pro quite cheaply on ebay.