Google Sheets count how many cells contain a specific word / words?

Not exactly answering what you are doing wrong, but here is what you can do:

=COUNTIF(A1:A100, "*WORDHERE*")

You don't need SUM around the COUNTIF.

=COUNTIF(A1:A100, "*WORDHERE*") 

will work just as fine. The same can indeed be achieved with regexmatch in a more complicated formula:

=sum(ArrayFormula(N(regexmatch(A7:A, "WORDHERE"))))

Here N-function is used to 'convert' the boolean values (TRUE or FALSE) to 1 or 0.