For best VO2 Max accuracy should I enter my morning body weight or my actual running weight with boots, coat, and leg and arm weights?

Just choose one option and keep at it to monitor changes.

VO₂ max with weights of course will differ from a standard one, same as running in heat or on harder terrain.

But even if it drops because of such conditions, then after few normal runs it returns to higher value.

Ideally I'd like my VO2 Max number to be consistent and increasing of course.   I am willing to adjust my weight values before a run to match my actual, kitted out weight.   It would be interesting to see if that VO2 Max reading is the same as if I put in my birthday suit weight and ran naked through the woods.  Since I live in civilization that particular test will never be  run.  :-)

Running shorts are quite light, and usually only those are required in public, so a test would be close enough.

Speed seems to be most important for VO₂, so not so easy to always increase it. When I do few sightsee runs in a row, then it drops. But after few effort trainings it goes up again, so no worries when it temporary drops.

And usually it's not updated just after one run, rather few changed runs are needed for it to drop or climb. If you do like 2 easy runs and others with effort during week, then it stays constant.

To check overall fitness, I would just look at max VO₂ during a year.

I checked to be sure when it updates and yes, usually it stays constant if you train and mixing some easy runs. But when you slow down during some consecutive activities, then it drops. And then if you noticeably speed up, it updates just after one activity.

For example you can drop to 48 and get to 50 with one fast longer run.

So basically if your training is consistent, then VO₂ is too, if you suddenly change speed, then VO₂ changes too.

About running with shorts only, it's a bit easier, but also it's done during summer, so it's harder because of heat. So it won't really give better VO₂ score. You can get best results when it's slightly cold as water is not so needed. I mean when running alone without support stands (drinks).