I have a vague recollection reading about a Russian version of the 9,3x53R, too. This is not the same as the Finnish version; seldom seem today but still loaded by Sako.

The bullet Sako used was 350grs. I think it was the Hammerhead, although I cannot vouch for this.
The Gamehead, I believe, is a later load: ammo at Lahdeniemi (scroll almost all the way down).

The 9,3x53R Finnish was a development in the 50's due to the then-legislation which prohibited the use of military ammo, i.e. 7,62x53R, by civilians; and requiring a minimum of 8mm bullet for moose hunt.
Thus, the military case was necked up to accept the common German 8mm (Mauser) and 9,3mm bullets.
The 8,2x53R is still often seen today but the 9,3mm I have seen seldom, indeed. The 9,3x53R was nicknamed "hirvensurma", 'death of moose'.
Most commonly the military "Pystykorva" rifles were converted but also a large number of Winchester model 1895 lever action rifles were converted for both cartridges.

- Lars/Finland