You've lost me a bit there Ken. What is a "drawbar"? I was unaware that you could have a threaded open end on the arbor. What is the purpose of it? I'm trying to learn as much as possible about turning, but I'm afraid I still have lots to learn. All I know is that I've been bitten by the turning bug and the itching won't go away!
A drawbar is a threaded rod that passed through the headstock spindle and screws into the MT fixing.
In many instances, the MT fixing is one end of a 'between centers' mounting arrangement in which the taper cannot possibly come out of the spindle while it is spinning. But there are a few instances (such as using an MT jacobs chuck in the headstock) where the piece being turned in held in the jacobs chuck alone, and the tailstock isn't used. In that situation, it is possible for the taper to work its way out of the spindle, and if that happens while you are turning, it can be dangerous.
So a drawbar is just a rod that screws into a female thread in the small end of the morse taper, and with a knob on the other end that can be tightened to keep the taper firmly seated inside the headstock spindle.
Tapers that are designed for use in the tailstock typically don't have the female thread since tailstock tapers generally aren't hollow and can't use a drawbar. Furthermore, anytime the tailstock is used, the workpiece is between centers so there is no chance of the taper coming loose.
By the way, there is a clear specification for morse tapers that defines the length of the tapered portion, and also the angle of the taper. But, occasionally you will encounter a 'short taper' - a fixing with a morse taper shaft where the angle of the taper meets the standard specification, but the length is shorter than standard. Fortunately, this is relatively rare and I suspect will only be the case for some old accessories - I've never seen a 'short taper' on a new accessory. This isn't a problem if the fixing is used in the headstock since you will typically use a knockout bar to remove it. But it can be a problem if you use a short taper in the tailstock because it may not engage the auto-eject feature where the taper pops out as you withdraw the tailstock ram.