I am sure they could decide to call themselves whatever they wanted. In some places calling a Social Democrat a "Marxist" would be fightin' words.
What names political movements embrace depend on the time (what century, what decade) and the place (what country) and the current political environment.
Lots of names chosen in the past are not instructive as to political decisions today.
The Marxist split occurred due to ideological differences among followers of Karl Marx, primarily revolving around differing interpretations of his teachings and tactics for achieving a socialist society. This led to factions such as the reformist Social Democrats and the revolutionary Communists diverging into separate movements.
Absolutely not. Hitler was a National Socialist, which is a right wing party. His main opponents were the Social Democrats.
There are no Marxists in the US government. There may be a couple Social Democrats here and there, but no Marxists.
Social Democrats were characterized as "the extreme wing of the socialists" who were "inclined to lay so much stress on equality of enjoyment, regardless of the value of one's labor, that they might, perhaps, more properly be called communists."
The Socialist Revolutionary Party, a non-Marxist party was formed in 1901. It had two factions, one for political agitation and one for terroristic violence. This party had nothing to do with the Bolshevik Party, which emerged from a split in the Russian Social Democratic Labor Party, which was a Marxist party.
Social democrats of which country?
Party of Social Democrats was created in 1994.
Åland Social Democrats was created in 1971.
Social Democrats of Croatia ended in 1994.
Social Democrats of Croatia was created in 1990.
Gibraltar Social Democrats was created in 1989.
Croatian Social Democrats was created in 2004.