I mean Marx used the term "communism" in his writings and back in 1848 he did to some extent self-identify as a communist, but it was more of a general "vibe" than a coherent political project. Communism as an organized, international political movement only structured itself in the 1920s as a product of WW1 and the Russian Revolution. So while it isn't exactly incorrect to call Marx a communists, it is genuinely historically misleading.
Marx's most direct political heirs are indeed the social-democratic and labor parties that emerged across continental Europe (UK Labour is a slightly different beast). These parties were directly inspired by Marxist theories and their organizational structure and political strategy was directly modeled on Marxist understandings of society. Of course these parties eventually deviated from Marxist ideology in important ways (although this happened later and less dramatically than is often assumed), but here's the thing: so did Lenin and the Bolsheviks who would come to structure 20th century communism. The Bolsheviks' theories and actions from around 1905 onward directly contradicted almost every orthodox Marxist principle, and as for the USSR itself, Marx would have very little good to say about it.
So I guess saying "Marx was a communist" and "Marx was a social democrat" are both true from a certain point of view, but both are also deeply misleading and are best avoided.
If this is what Laki is getting at then I don't have any major disagreements. The italicized is something I really wish a lot of Leninists would come to understand.