Hi Heather! (and Teemu, if you are still around).
This is just an opinion: I think the first man in this line should be Martti Marttinen at birth, because he was born in Finland (even if the priest or taxman was Swedish and wrote the name down in Swedish or Latin). About the existence or nonexistence of birth records in Finland in the first decades of the 1600s I know nothing.
His namesake son should be Mårten Mårtensson at birth, because he was born in Sweden (in the province of Värmland, presumably, because the Finns that went on the Eagle in 1654 were recruited from Värmland). Then they should probably both have their names in all the three languages somewhere in their profiles.
"The Swedish Colonial Society uses Mårten Mårtensson" - and, I suppose that Swedish genealogy does the same, because Swedish genealogy in general will rather use the principle of having standardized/agreed-upon names for individuals and families (often stating the principle for standardization in the intro to a given work) than the WikiTree principle of basing profiles on their Last Name at Birth.
About the existence of early birth records in Värmland I do know something: there aren't any. There are very few parishes in Sweden where there are birth records for, say, the first half of the 1600s - which is, of course, unfortunate for all the emigrants to New Sweden. You need to be a real specialist/historian to know what other types of sources there may be and how to find them :-(
