Ok, so after watching the video and doing more tests/measurements this is all making a lot more sense to me now.. I should have included what I was measuring in my first post. Thank you guys for not laughing at me too hard trying to figure this out, haha.
I was measuring the current of a flashlight at a set mode/level, 18650 cell around 4.15V. The main thing I was trying to figure out was why I was getting such different readings with the new meters vs the older ones. The answer is exactly what you guys said - the 'Burden Voltage'. After learning this is a thing, now I know that the reason my vintage flukes all read close to the same (but different than the newer ones) is that they all have a burden voltage very similar to each other. The same for the newer meters, they are pretty close to the same burden voltage - quite a bit lower than the older ones.
Apparently, what I was measuring was pretty sensitive to the burden voltage load in the circuit which translated to voltage drop hence the mA increase etc.. (if I am thinking about all that correctly).
The suggestion to measure it with the meters in series was a good idea, when I did that they measured very close to the same. Just a different reading than I was expecting (newbie me)...
I presume that if I was using a power source that maintained a set voltage under load I would get a pretty similar reading with the new and the old meters, or at least a lot closer than the readings I am getting now?
Thank you all for your help! I've learned a lot, hopefully someone else can learn from this also. That is part of the fun right?