Like Pivi says, both cases have a special anneal for forming.
I have no idea which came first. From the appearances of the boxes I would guess that they were introduced at the same time. Only a guess though because my memory of those days is not that good anymore. The boxes for each are the same externally. I do know from experience that the 22BR cases are not nearly as common as the others and are collectable. The other headstamp is fairly common (I have several full boxes of each).
Just so collectors know, cases can be found with and without flasholes, with both a small and large primer pocket, and a few with no headstamp. These variations are very uncommon.
This little experiment by Remington was one of many boondoggles on their part and cost them the short-range benchrest market. Remington actually made a 6mmBR Benchrest rifle for a short time and since Benchrest shooters are wildcatters and handloaders, Remington, for some reason, thought that shooters would like to make their own brass for the 3 BR cartridges - the 22, 6mm, and 7mm. For a very short period of time a few actually went that route. But as soon as LAPUA 220 Russian brass became available the BR was dead and the 6PPC became king.
Remington saw the error of their ways and finally made a 7mmBR and 6mmBR factory case but it was too late. When the 6mm BR started making waves as a mid and long range Benchrest cartridge LAPUA once again pre-empted them with their outstanding high-quality brass, so even the properly formed and headstamped 6mmBR and 7mmBR brass is popular only with those shooters wanting to save a few bucks on brass. Even that is penny wise and pound foolish because the LAPUA brass lasts a lot longer. Collectors probably should hoard a box or two of Remington brass because I can’t believe it can survive much longer.
LAPUA once again saw a future market when they introduced the 6.5x284 and 6x47 a few years ago. Those two cases are runaway best sellers. Either Remchester has an ignorant marketing staff or they don’t care. Probably both.
Pivi is correct that the BR series is a descendant of the Frank Barnes 308x1.5" wildcat of the 1960’s.