A value is being populated in the next column because you are effectively creating a 2-column array with the first argument of the IF.
For an array formula you can use the + operator to apply OR logic:
=ARRAYFORMULA(IF(LEN(A3:A),IF((C3:C="Name1")+(C3:C="Name2"),"A",""),))
or if you had a longer list of allowable entries, something like this would be easier to scale:
=ARRAYFORMULA(IF(LEN(A3:A),IF(ISNUMBER(MATCH(C3:C,{"Name1";"Name2"},0)),"A",""),))