Again, the following code, using a regex from postings elsewhere in this thread, always perfectly defines  $2 (although sometimes it is defined as the empty string – perfectly valid!) and seems to give the required output:

>perl -wMstrict -le
"my %words = map { $_ => 1 } qw($Word $name $test);
 my $line = '$Word $ $ $ $test $testing';
 $line =~ s/(\$(\w*))/exists $words{$1} ? $1 : qq{dollar$2}/eg;
 print qq{$line};
"
$Word dollar dollar dollar $test dollartesting
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Note that  (\$(\w*)) is the regex used.