In your video, it sounds like it's cranking over fine, especially for a first, cold start with 2 glow plug cycles. Even my Infiniti gasser turns over a bit slower an longer on a 30 degree or lower cold start. I think if you would have cranked for a second or two more on the first attempt, it probably would have started. That extra couple of seconds of cranking wouldn't have hurt the starter.
If it keeps starting like this, I wouldn't be worried about the starter or batteries. Sounds like both of my IDI's.
I do understand being conservative thought, with only having one vehicle. I have the luxury at this point, that I have more than one vehicle, and if one doesn't start, I just jump in another one. I have seen batteries pass on a "tester", but don't have the oomph to start a diesel. A diesel, with running the plugs, and then cranking against the compression, the batteries just don't have what it takes. I know on my 6.0, which is a power hog, if the batteries are low and it will turn over slowly, if I try and jump it, it takes a small charger on 75 amp boost and cables from a running vehicle, or 2 sets of cables from another diesel or 2 vehicles. A diesel takes a tone of power to crank over.