The reason for the observed behaviour is that a regex with "s as the delimiter get interpolated as a string, before it gets treated as a regex.

I think that the docs are simply out of date or perhaps somewhat badly phrased. Any expression on the right-hand side of =~ or !~ is treated as a regex.

print 'String contains a zero' if '0' =~ 0;
String contains a zero

$_ =~ join '|', 0, 1 and print "'$_' contains a '0' or a '1'" 
    for qw[ 0 2 50 02 1 11 abc1 def];

'0' contains a '0' or a '1'
'50' contains a '0' or a '1'
'02' contains a '0' or a '1'
'1' contains a '0' or a '1'
'11' contains a '0' or a '1'
'abc1' contains a '0' or a '1'

$_ !~ join '|', 0, 1 and print "'$_' doesn't contains a '0' or a '1'" 
    for qw[ 0 2 50 02 1 11 abc1 def];

'2' doesn't contains a '0' or a '1'
'def' doesn't contains a '0' or a '1'
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The only time the 'm' or '/'s are required is when implicitly comparing against $_.