Your problem is that by measuring “probe to probe” you are actually measuring the voltage drop across your test leads, as well as the meter.
Thus proving how cr@p your leads are. And I reckon one of your two sets is more cr@p than the other. Whereas the meters themselves are probably a small fraction of the overall drop. Swap the leads around and see.
Trying to measure the resistance of one meter with the other is futile. First of all you are probably doing it through the dodgy leads, which will dominate, secondly the actual resistance will likely be a few tens of milliohms, which your meter cannot resolve, even to one count.
You are probably just measuring the resistance of your leads.
Your meter may be a basic one with a minimum scale of 200 ohms. And a resolution of 2000 counts (i.e. 0.1 ohms). A better one might have 22,000 counts (4 digits). Even so, wrong tool for the job.
Half a volt loss is massive. Its like using a battery at 3.7 volts instead of fully charged at 4.2 volts.
To measure the actual resistance of the meter measuring the current, you need to measure the voltage across it at the meter terminals.
Difficult with shrouded leads, but if you can trap a strand of fine wire into the plugs you can monitor it there.
Of course with my plugs it is easy to connect straight onto the gold body.
With the meter I mentioned (£5) the total resistance through the two plugs and the meter was a mere 15 milliohms. (Volts drop divided by current).
Which is plausible, if you look inside you’ll find that the current shunt is just a piece of stout tinned copper wire. Sometimes just a thick PCB trace. No fuse etc. Very robust.
That’s also the reason they are not continuously rated. E.g. maximum 15 seconds, then rest for several minuutes.
It’s not that they can’t handle the current continuously, its because the copper shunt used increases in resistance as it warms up. About 0.4% per degree C.
As will your test leads, but more so.
A more expensive meter might use a better internal shunt, with much less temperature variation.
As the resistance increases, the current measurement becomes over-optimistic, i.e. a 10 degree temperature rise would raise the reading by 4%
I don’t usually worry about that.
These cheap meters are actually quite useful for tailcap measurements, providing you use them correctly (short stout leads). Despite the FUD spread by others on this forum who don’t necessarily understand what’s going on, or who prefer to spend much more money on “better” meters, but don’t know how to use them properly either.
Try measuring the volt drop at the meter terminals and let us know. I think you might be pleasantly surprised. 
And knock up a set of short stout leads, I’m sure you will see the difference.
Resistance is futile… :bigsmile: