What I have to say is specifically for gas grills, as I have baked many dozens (or hundreds) of NY style pizzas in gas grills. Although it appears to me that your grill is a charcoal grill, my information might still be helpful. (I don't know, though, because I've never baked in a charcoal grill.)
- If the stone gets too hot and burns the bottom of your crust before the top of the pizza is done (which happens every time in unmodified gas grills), line the bottom of your stone with aluminum foil. Weber's recommendations, as mentioned in your post, may accomplish the same goal, but I'm not sure.
- Try not to open the lid while baking, to keep from losing all your top heat, because you'll probably never get as much top heat as you'd like.
- Try to figure out how long it takes to bake a pizza without opening the lid, and set a timer every time you bake a pizza once you figure out the typical bake time. For me this is about 4 minutes, but it may be a little less than 4 minutes if the grill is hotter than usual.
The last two things I mentioned take some trial and error to figure out. So maybe start by setting a timer for 3 minutes, then give the pizza a look after the alarm sounds. If that doesn't do it, then try 4 minutes the next time. If that's not enough time, try 5 minutes.
My tips are for pizzas that you peel onto the stone. If you use a screen or a pan, that changes everything.
Also, with the last couple pizzas I baked in the grill, I've come to prefer a 5-minute bake at a slightly lower temperature. I can't tell you the exact temperature because I don't have an IR thermometer and because the lid thermometer reading is clearly not representative of the stone temperature or the temperature around the stone.
Hope that helps.