Using function parameters to improve testability

I'm interested in finding out which is a better way to do this. Say I have the following object

var myObject = (function(){
    function myObject(myName){
        this.myName = myName
    };

    myObject.prototype.checkNameIsJames = function(){
        return this.myName === 'James';
    }

    return myObject;
};

var newObject = new myObject('James');

To test the checkNameIsJames method, I rely on a parameter passed into the constructor. This works nicely because everywhere I can call it without parameters:

var result = newObject.checkNameIsJames(). 

But it is always bound (in this specific situation) to the constructor. The other way would be to supply the name as a parameter:

var myObject = (function(){
    function myObject(myName){
        this.myName = myName
    };

    myObject.prototype.checkNameIsJames = function(name){
      return name === 'James';
    }

    return myObject;
};

var newObject = new myObject('James');

In which case I can do all kinds of unit tests with different names

var o = new myObject('James');
var result = o.checkNameIsJames(o.myName);
var result = o.checkNameIsJames('Peter');

This is a fairly contrived example, but if you imagined this as a much larger example with many objects functions and vairables, should I be aiming to use parameters as much as possible even when not technically necessary so that I can improve the coverage of my testing? Or would that be considered uneeded complexity for the sake of completing unit tests?