It looks to me like you don't care if the key is defined, what you're interested in is if there's a value. So I'm not bothering with defined or exists here:

$c = 0; for (keys %hash) { $c++ if $hash{$_}{'x'} && !$hash{$_}{'y'}; 
+} print "$c\n";
$c = 0; for (keys %hash) { $c++ if $hash{$_}{'z'}; } print "$c\n";
$c = 0; for (keys %hash) { $c++ if $hash{$_}{'z'} == 101; } print "$c\
+n";
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