Finland's border were unchanged by independence in 1917. That is, the borders with Norway and Sweden remained the same, and the border with Russia was the old border between the Grand Duchy of Finland and Russia, when Finland was part of the Russian Empire. This border mostly dated to 1812, when the parts of Finland that had been gained from Sweden by Russia in 1721 and 1743 were joined with the rest of Finland, obtained from Sweden in 1809. In 1812, Sestroretsk (Siestarjoki in Finnish) also became part of Finland; this town had been founded by Peter the Great in 1714 around an arms factory. In 1813, the border in Ostrobothnia and Lapland was moved east. In 1842, part of Siestarjoki returned to Russia, and in 1864, the rest of the town returned to Russia. In return, the czar promised Petsamo to Finland, giving Finland access to the Arctic Ocean.

There were attempts to shift the border, by Finland, via intervention in Karelia, Ingria, and Estonia (the "Heimosodat", or "Kinship Wars"), and an attempt to take Petsamo, during the Russian Civil War. These were ended by the Treaty of Tartu in 1920, in which Petsamo was handed over to Finland, and otherwise the border remained the same (which meant Finland gave up control over Repola and Porajärvi, two regions in Russian Karelia adjacent to Finland which had voted to join Finland).

The next, and last, changes in the borders were due to the Winter War and Continuation War, with Petsamo returned to the Soviet Union, and Finland losing Finnish Karelia and part of Salla: https://commons.wikimedia.org/wiki/File:Finnish_areas_ceded_in_1944.png

Petsamo was the eastern "arm" in Lapland - the Lapland "map man"was one-armed from 1917 to 1920 (and again after 1944). The western "arm", bounded on the west by the Muonio and Tornio rivers, was the border between Sweden and the Russian Empire resulting from the Treaty of Fredrikshamn in 1809. (The upper reaches of the Muonio is called the Könkämäeno.) The western side of the "head" is mostly the Tenojoki (Tana) - this is part of the border with Norway. Apart from the Teno/Tana river, the Finnish-Norwegian border largely follows the watershed to the north and east of the the Könkämäeno. Thus, the western "arm" and the "head" mostly result from the border with Norway, which the Grand Duchy of Finland (i.e., Russian Finland) inherited from the old Swedish-Norwegian border.

Considerations about giving the Lapland "map man" a second arm don't appear to have been in any way a feature of the discussion about giving Petsamo to Finland in 1864 (in exchange for Sestroretsk/Siestarjoki). In the event, it took Finland until 1920 before it finally obtained Petsamo. The western side of the Petsamo "arm" was determined by the previous Russian-Norwegian border, and the eastern side was a very modern straight line drawn on the map.

Map showing the topography of the Finnish borders with Sweden and Norway: https://commons.wikimedia.org/wiki/File:Map_of_Finland-en.svg