Was the word “communist” used prior to Marxist/Leninist writings?

Solution 1:

The word was used in English in 1840 according to the OED, which I believe narrowly beats out Marx’s writings:

Noun:

The Communists have their meetings, and the Radical Reformers, who do not go the length of an agrarian law, dine together in numbers.

1840   Morning Chron. 13 July 2/7 

Adjective:

A social banquet of the adherents of the Communist, or Communitarian school is expected to take place.

1840   J. G. Barmby in New Moral World 1 Aug. 75/1

Communism:

A man named Dufraisse..concluded with an exposition of the doctrines of Communism..much the same as what Mr. Owen preaches in England, under the name of Socialism.

1840   N.-Y. Spectator 22 Aug. 2/1 

Both communist and communism in this sense come from French.

Solution 2:

communism ~ 1843 and communist ~ 1841

Etymology and 1st Usages:

communism (n.)

1843, "social system based on collective ownership," from French communisme (c. 1840), from commun (Old French comun "common, general, free, open, public;" see common (adj.)) + -isme (see -ism).

Originally a description of a society, by the early 20c. it was a general a term of abuse for revolutionaries, implying anti-social criminality without regard to political theory. The full etymology in in the citation.

The noun communist now has the following definition: TFD

a. A member of a movement or political party that advocates Communism. b. A supporter of such a party or movement. 2. A Communard. 3. often communist A radical viewed as a subversive or revolutionary.

So to answer your question:

• But was the word 'communist' used prior to Marxist writings? Or was it coined for use in the Communist Manifesto?

The were very closely co-located in time. It appears the word was incorporated into the English corpus (~ 1841) just prior to the Communist Manifesto (1850).