external JavaScript function not triggered in HTML <script>

I know this question has been already addressed here but that workaround does not work in my case.

I have a function in an external JavaScript that I would like to call immediately after the page is loaded. This is the snippet in the external code to be executed

var user_id = "";
var obj = obj || {};

obj.Id = {
    assignId : function(id) {
        console.log(id);
        user_id = id;
        window.alert("ID no:" + user_id);
        return user_id;
    }
}

and this is the HTML

<script>
obj.Id.assignId("a string");
</script>

I also tried this way

<script>
window.onload = function() {obj.Id.assignId("a string");}
</script>

but window.onload is called afterwards in the external file so I think it was overridden as described in this question. Here is what I get from the console in both previous cases:

Uncaught ReferenceError: obj is not defined

When using this way:

window.onload = function() {
    obj.Id.assignId("UA-0001");

}

I don't get any error but the function is not triggered though.

What I'm not able to figure out is: if prompting directly in the console obj I get Object {Id: Object} and if prompting obj.Id.assignId("a string") I get exactly what I want.

Is there any explanation for this?