Whenever I have seen North American 3 coil high pressure sodium ballasts that are typically found in GE high pressure sodium lanterns, I often see these ballasts being referred to as mag-reg and regulated lag and they have a dedicated coil that is used with a capacitor. I wonder if these 3 coil ballasts function like HX autotransformer ballasts if they are considered "lag" type autotransformer ballasts and have the tendency to run lamps with an OCV requirement that is twice the mains voltage. In addition, I wonder why these particular ballasts use a dedicated coil that is specifically designed for capacitors.

Here is what they look like:

https://www.lighting-gallery.net/gallery/displayimage.php?album=307&pos=163&pid=190408

No.
They are really referred as "MAGnetically REGulated lag" ballasts.
HX transformers are just the single primary/single secondary transformers with designed in magnetic leakage path. In HX is no core saturation based voltage reguulation (like the mag-reg; the regulation happens from the first to the 2'nd coil; the 2'nd-3'rd is then the HX like ballasting), nor current regulation (like in CWA, where the saturating shunt detunes the series LC ballasting impedance).

No.
They are really referred as "MAGnetically REGulated lag" ballasts.
HX transformers are just the single primary/single secondary transformers with designed in magnetic leakage path. In HX is no core saturation based voltage reguulation (like the mag-reg; the regulation happens from the first to the 2'nd coil; the 2'nd-3'rd is then the HX like ballasting), nor current regulation (like in CWA, where the saturating shunt detunes the series LC ballasting impedance).

I wonder if these ballasts behave like HX ballasts or CWA ballasts.

I wonder if these ballasts behave like HX ballasts or CWA ballasts.

Except their size and losses, they combine advantage of both:
Load characteristic close to the series choke/HX (so with increasing load voltage the current is sloping down, so suppressing the thermal instability of saturated vapor lamps), suppression of the mains variation (actually better than CWA use to; CWA is able to cover typical +/-10%, in extreme designs up to +/-15%, the mag-reg up to +/-30% feed supply variation).
Plus it could be easily made with an isolated secondary without that much of a cost or efficiency penalty.
But the "price to pay" is the larger mass and cost and higher ballast losses.

So because of that, their primary applications was highway HPS lighting, where long wiring made large voltage drops (the voltage stabilization suppresses that) and where the lamps are of saturated vapor type (HPS), even when the US market HPS are designed with the thermal feedback suppressed to some extend. Plus the isolated secondary allows the lamp shell to be kept grounded regardless how the input is connected (e.g. across phases,...)

I am also starting to question whether the mercury vapor and probe start metal halide regulated lag ballasts can properly run saturated vapor high pressure sodium retrofit lamps such as the Sylvania Unalux lamps since I have read in the documentation that says that they must never be used with CW and CWA ballasts and only on simple choke (reactor) and HX autotransformer (lag) ballasts, but the documentation makes no mention of regulated lag ballasts.

I'm afraid there is no general guarantee. It will depend on how the particular ballast is designed. Because there is no formal standardization, the universal compatibility can not be guaranteed.
But the thing is, in the age of electronic ballasts (with inherent 1000% better mains variation suppression) and way more efficient light sources, no one would invest in evaluating the compatibility of such inefficient ballast anymore.

Plus I'm not even sure, if the regulated lag ballasts for MV really existed: CWA is way simpler, cheaper and more efficient, yet still able to handle very wide mains voltage variation and well compatible with the MV, as well as probe-MH. It were the requirements of HPS and modern saturated vapor MH, what required certain ballast characteristics, which can be attained together with the wide mains voltage tolerance only with the regulated lag style, so the expense in purchase cost and ballast losses was seen as still acceptable price to pay. Of course before electronic ballasts took over.