If "=~ is the pattern applying operator", then code like if( /foo/ ) would never apply the regex. The two sides of =~ are *not* passed to some built-in function as arguments. The right-hand side is what is run and the left-hand side is the argument if it is present. An "unbound" regex still gets applied. Matching happens even without =~.
The =~ (mostly) is *not* what makes pattern matching happen. The regex is what makes pattern matching happen and the =~ (if present) binds a string to the regex to tell it what to match against (other than the default).
Now, if you have =~ with something that isn't a regex to its right, then you have rather bizarre code and are at the mercy of dubious "do what I mean" guessing. If Perl happens to currently insert a regex constructor for you, my main reaction is that I know I have no code that relies on such.
So the reason we say 'bind' is because one side is the code to run. Like '(' is used to bind arguments to a function -- '(' is not "the function invoking operator", despite that description being somewhat appropriate.
In reply to Re^4: Why do we say the =~ operator "binds"? (unapplied?)