"To test env -S on the command line, use single quotes for the -S string to emulate a single parameter. Single quotes are not needed when using env -S in a shebang line on the first line of a script (the operating system already treats it as one argument)"(from your second link).

This is different for shebang on Mac though:

GNU env works with or without '-S':

#!/opt/homebrew/bin/genv -S bash -v

echo "hello world!"

BSD env works with or without '-S' too:

#!/usr/bin/env -S bash -v

echo "hello world!"

To conclude, looks like adding `-S` is the safest option for comparability sake :).