[SalesForce] Javascript function gets incorrect boolean value from controller

You can change up some of your visualforce and javascript slightly to achieve this.

First lets change up the JS a tiny bit so it directly references the checkIn apex method and is wrapped in an element you can rerender.

<apex:outputPanel id="scriptContainer">
    <script>
        //on dom ready code is fine
        function backToDetail() {
            alert('{!checkIn}');
        }            
    </script>
</apex:outputPanel>

Now just change up the actionFunction definition so it doesnt call checkIn and instead it rerenders the script tag containing the backToDetail method. The purpose of doing this is that when it is doing the rerender it will replace alert('{!checkIn}') with alert('Yes') or alert('No').

<apex:actionFunction name="checkIn" oncomplete="backToDetail();" rerender="scriptContainer" />

You could wrap a script in an outputPanel and rerender it on every save. Then you'd have the most up-to-date value of the variable. eg:

<apex:outputPanel id="saveResultScript">
    <script>
    function confirmProceed (){
        var dataSaved = "{!JSENCODE(dataSaved)}";
        if(dataSaved == 'false' && !isDirty) {
            if(confirm('Are you sure you wish to continue - any unsaved changes will be lost.')) {
            }
        }else {
            return false;
        }
    }
    </script>
    </apex:outputText>
</apex:outputPanel>

Outside this block we have another script tag which we'll put the isDirty variable (so it doesn't get rerendered on save)

<script>
var isDirty = false;
</script>

As long as this entire panel is rerendered on every save, we can check the newly updated dataSaved variable and the locally updated isDirty variable (you said you had a variable that performs this function on any onenter event)

This way, we don't need to talk to the controller for confirmProceed to work. Hopefully this helps a little. I feel a little unsure of your requirements, but this is something that I have done several times myself and it works pretty well.

Oh and using a RemoteAction as suggested in the comments would probably work too, as long as you keep in mind that RemoteAction methods need to be static. They definitely return a value immediately for you to act on.

Update One last thought - you could use a RemoteAction to save your data and then (depending on the result) call an actionFunction to update the dataSaved var on the controller non-statically - while you can do what you want without waiting for a rerender - this should be very performant.