If the drawing merely says "M2", "M3" or "M4", then take is as read that it s a Metric Coarse thread.
Don't worry about the Fine and Extra Fine threads unless specifically called for. They may be nice to have, but won't see much use, if any!
I have sets Taps and Dies for different threads; some of which have never been used.
(Years ago, at work, I dealt with UNEF threads, but have never found the need for them at home, so have no facility to produce them, other than setting up then lathe if ever needed).
M.E.40 threads are useful, as a fine adjustment, (0.025 inch / rev for Luddites like me).
Russel's suggestion of gripping the work in a split collet is a very good one, and will prevent the tube being crushed (At 3/16 OD it likely to be fairly weak, especially if thin walled)
To really insult you, you are offering the end of the die with the larger internal chamfer onto the work, aren't you?
It SHOULD be the end marked with the thread details. I have been supplied with the odd die where the marking was, incorrectly, on the other end, but Tracy changed it without any demur.
Have you a female thread, into which this 3/16 brass tube has to be screwed?
If so, my suggestion is to use this as a gauge. Back off the outer two screws in the die holder, and tighten the centre one, to open the die, and then cut a thread. Try the result in the female. If it won't even start, then slacken the centre screw a little, ( maybe barely a quarter turn) and run the die down the thread again, before retrying. Keep on doing this until you get a thread fit which feels right, (smooth without being loose), but remember that a small adjustment can make quite a difference. If it squeals, it's too tight!
As a final check, then cut a thread on "virgin" tube, and check that the fit is as you want. If it is, you are ready to go into production.
On the subject of lubrication, it is normal not to lubricate copper or brass, because of their property of acting as a low friction bearing material. Cast Iron contains graphite, so that acts as a lubricant, and so no extra is needed. For Aluminium, something like kerosene (paraffin) is the usual recommendation. For steel, I use either Rocol STD, or quite often, the fat melted off the breakfast bacon! After all, Tallow was used by engineers before the advent of more advanced lubricants.
Hope that this is some help, but all these posts show how Model Engineers are always prepared use their experience to help each other, in whatever way possible.
Howard