Do electric eels produce current as they just move about, and how/why?
You are right - the 800 Watts are not constantly produced:
There are three types of electrical discharge: (i) low voltage (ii) high voltage doublets or triplets and (iii) high voltage high frequency.
(i) low voltage allows the animal to orient itself in its environment [b].
(ii) high voltage doublets or triplets are used for hunting: By discharging a high voltage doublet or triplet the electric eel induces involuntary twitching in the prey animal. Since this twitching is producing a weak electric field, this allows the electric eel to locate its prey. See Figure 3 of [b].
(iii) high voltage high frequency is used to incapacitate the prey. It's about 400Hz at up to 600V [b]. Wikipedia [c] states that these electric eel can produce about 1Amp of current. That's in the region of 800Watts the narrator is speaking of. Edit2: The pulse duration is about 1ms [e]. With a frequency of 400-500Hz this leads to a surprisingly high ~50% dutycycle. The ability to keep the frequency stable depends on the age (=length) of your eel: The older the eel, the more stable the inter-pulse interval.

Caption: Fig3 from https://doi.org/10.3389/fnint.2019.00023, red=50cm, green=75cm, blue=115cm length of eel
In Figure 5 of [a] you can see that the electrical eel is able capture prey in ~5 sec. During these 5 seconds the electric eel is pretty much constantly producing high voltage discharges. These could be used to light a Christmas tree.
To summarize: Yes - electric eels are very much in control of when and how they discharge. This can be quite clearly seen (or rather heard) in [d] which is  Supplementary Movie 8 from [a]
[a] https://www.nature.com/articles/ncomms9638
[b] https://s3.amazonaws.com/vu-wp0/wp-content/uploads/sites/182/2015/11/Science.pdf
[c] https://en.wikipedia.org/wiki/Electric_eel
[d] https://static-content.springer.com/esm/art%3A10.1038%2Fncomms9638/MediaObjects/41467_2015_BFncomms9638_MOESM559_ESM.mov
[e] https://www.frontiersin.org/articles/10.3389/fnint.2019.00023/full