the only difference is that variable being the defaulted $_ while the other is user-defined

Nope, no "user-defined" variables being used by s/// in either snippet.

my $r = "Time to work";
s/work/play/;
print $r;
[download]

means

my $r = "Time to work";
$_ =~ s/work/play/;
print $r;
[download]

Since you never assigned anything to $_, you get a warning. Conversely, you never told s/// to work on $r. Doing either will solve the problem.

my $_ = "Time to work";
s/work/play/;
print $_;
[download]

my $r = "Time to work";
$r =~ s/work/play/;
print $r;
[download]