I was told that if you buy a lower it is sold and transferred as "Other
Firearm," thus allowing you to build it into anything you'd like.
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This is correct. Either a stripped or completed lower, sold without a barreled upper, is considered an "other" firearm and can be assembled into any legal config, including a pistol. There is no requirement for it to be engraved "pistol". The key here is what the
first,
complete assembled configuration of the firearm with that receiver is which will determine what it is.
If it is first completed as a pistol (no stock, and barreled upper attached of any barrel length), then it is a pistol. As a pistol it can be changed at any time to a rifle by first adding a barrel >16", and then adding a stock, and likewise you can then reverse that and return it to pistol config at any time as well.
If it is first completed as a rifle (stock, and barreled upper with barrel over 16" long), then it is a rifle. If it starts out this way as a rifle
first, then by law it is always a rifle and can never legally be reconfigured into a pistol.
Here in Ohio, there are no specific laws regarding AR type pistols. No barrel length restrictions or anything. The only thing is the still currently active law preventing use of magazines that hold over 31 rds, but luckily in less than a month that law will no longer be in effect.
Also, current interpretation on the use of your CHL with an AR pistol it that yes, it does work and as such you can conceal carry one should you choose, but obviously the more practical application of this, as you pointed out, is the ability to have it loaded and ready at hand in a vehicle.
There is a specific section of this site that deals specifically with AR pistols
HERE, and should help you with further research on them.