8x57J - NOT the old Gew.88 bore. A derived standard for civil rifles only. See later correspondence. Note the use of J, not I, to avoid confusion with the number 1.
Bore / Groove 7.80 / 8.07 mm = 0.307 / 0.318

8x57JS - the "classic" Gew. 98 /Kar 98. S stands for "Spitzer" bullet, already mentioned in previous posts.
Bore / Groove 7.89 / 8.20 mm = 0.311 / 0.323

8x50R - the original Austrian Mannlicher miltary caliber. Most altered to 8x56R M30S
Bore / Groove 7.95 / 8.35 = 0.313 / 0.329

8x56R M89 Portuguese Kropatschek. Sometimes confused with the Hungarian 8x56 (below).
Bore / Groove 7.85 / 8.20 = 0.309 / 0.323

8x56R M30S - the later Hungarian M(odel 19)30 S(pitzer) caliber.
Bore / Groove 7.95 / 8.35 = 0.313 / 0.329

https://www.milsurps.com/showthread.php?t=60101&page=1

So a modern 244 gr .329 or .330 might be used per some references. Originally a lesser calibre projectile eg .324 was used, but the base somehow expanded to .329. ???

A .329 244 gr at approx 2000 fps.

The 8x50R was used as it feed through the magazine and no change to the bolt head. ???

Edited, I see you posted in between. Thanks.