I think your first example is backwards. Try

foreach my $sourceLine ( @SourceData )
{
    if ( $sourceLine =~ /\b$functionName\b/ )
    {
        $counter++;
    }
}
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Note that \b matches a "word boundary", which prevents matching on non-function names embedded in other "words" (including other function names). But it doesn't handle the problem where function names are mentioned in comments.