I only use distilled water on my sponge. I can't prove it makes your tips last longer....
Tap water can have minerals that will reduce tip life, such as sulfur. Distillation removes them = aids tip life, and as you mention, it's very inexpensive.
If you look at that guide, no They tell you not to do things that have been reported on here as being ok. For example, not using thin solder as there isn't enough flux and not to use anything harder than a soft wet sponge to clean the bits.
It's a good article overall, but I suspect it's a bit old.
Sponges used to be the only thing to use, and all that soldering stations came with. This has changed with lead-free solders though, which is when brass wool made it's appearance (less thermal shock than a damp sponge). Some stations even include brass wool these days (i.e. Hakko FX-888D has both wool and a sponge; suspect "sponge only" with a station is a cost cutting measure).
FWIW, you can always add more liquid or paste flux to the tip if the solder wire you have doesn't contain enough.
My new soldering iron tip is very solder phobic. (Solder beads up and pulls away from it when I touch it with solder wire.) How can I better keep my iron tip coated in solder?
Does the chrome plating go all the way down to the business/working end of the tip?
I ask, as solder doesn't wet to chrome (business end is supposed to be iron only, with only the upper part plated in chrome). Seen cheap tips on firesticks have chrome on all of the surface of the tip.
You could try some tinning compound, otherwise you may need a new tip (could sand it, but you run the risk of removing all of the iron plating, which will ruin it).