The last picture clearly shows the SNY cartouche (State of New York), not 8NY so yes, you have a New York State rolling block in 50-70. Tang markings indicate that this one was part of the first order of 15,000 made in 1871.
Another distinguishing feature of the New York rolling block is the longer hammer thumb piece than those found on other rolling blocks.
The following shamelessly plagiarized fromGeorge Layman's book on military rolling blocks "The New York State rolling block is also the only Remington made rolling block with the "locking action" which incorporates some internal modifications. It is loaded the same as the conventional rolling block action is, by cocking the hammer, rolling the block down and inserting the cartridge. However, when the block is raised the hammer automatically drops into a safety notch so that it must be returned to full-#### before it can be fired."