die() simply prints strings... false is the empty string, there you have automatic type conversion.
If you are still asking what is the problem with the code, it is still the same, unless you changed that...
You do not keep information about the fact the person logged in across requests. So the person logs in on a post, on the next get the $session is created anew, without it being logged in. In other words, your "Session" is not doing the thing that is the essence of being a Session.
Consider that, and try to figure out how you are going to deal with it. Usually people use $_SESSION to hold an identifier and check whether the person requesting the page is logged in already. That's really pretty much your only option.
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