That ".44 caliber rimfire" is most likely neither.
It looks identical to one in my own collection: a very early .45-70 cartridge made with an internal centerfire primer.
The giveaway is the cannelure just up from the base, this keeps the interior primer in place. The primer is placed into the bottom of the case, via the mouth, then crimped into place with the cannelure around it. Powder and bullet are then added.
Brief mention of this cartridge is given in the November/December 1997 issue of the American Rifleman magazine, page 19.
The AR notes that a variety of priming systems were experimented with from 1870 to 1880, most of it at Frankford Arsenal in Pennsylvania.
The inside priming system was in use in various forms until 1878. After that, the reloadable Boxer type primer was adopted. This is the (visible) priming system still in use today.
An excellent article on this cartridge, as well as other early, experimental and odd .45-70 cartridges appears in the 5th edition of Handloaders Digest, copyright 1970. The article is "Fodder for Forty-Fives."
This old book frequently appears on Ebay for a good price. It would be worth picking it up as a reference.