either the $k or the $closure method work

Be careful! Whilst the for my $name... works, it is dangerous:

my @names = qw[ the quick brown fox ]; 
undef %h; 
for my $alias ( @names ){ 
    $h{ $alias } = sub { 
        print $alias; 
        $alias = 'fred';      ####### Mysterious action at a distance 
+here
    }; 
}; 

pp \%h; 
{
  brown => sub { "???" },
  fox   => sub { "???" },
  quick => sub { "???" },
  the   => sub { "???" },
}

for my $key ( keys %h ) { $h{ $key }->(); };;
the
fox
brown
quick

print @names;;
fred fred fred fred
[download]

With the for my $name ( ... ) method, the closures are and remain aliases to the source list in the for loop; which means that if you assign to the closure within the subroutine, you will cause spooky action at a distance to the content of that source.

And perhaps worse, changes to the source of the for list, will remotely change the contents of the closures:

[0]{} Perl> 
my @names = qw[ the quick brown fox ]; 
undef %h; 
for my $alias ( @names ){ 
    $h{ $alias } = sub { 
        print $alias; 
    }; 
}; 
pp \%h; 
{
  brown => sub { "???" },
  fox   => sub { "???" },
  quick => sub { "???" },
  the   => sub { "???" },
}

$names[ 1 ] .= 'Mysterious changes';  ###### And again here!

for my $key ( keys %h ) { $h{ $key }->(); };;
the
fox
brown
quickMysterious changes
[download]