Client-side API design
After some testing I discovered that HomeActions._ajaxSettings is indeed static and would cause issues if simultaneous calls were made to the API. After some time away from the code  the obvious solution popped into my head as shown below. I've been too caught up in learning the confusing syntax of javascript that the basics of OOP design fell to the wayside just so I could get a working prototype.  

I added a "class" called Action and moved the Start() method to it and in each static method, RefreshCache and IndexAllRecords, I now return a new Action class which maintains its state separately with each call to the API.

var MVC;
(function (MVC) {
    var Action = (function () {
        function Action(ajs) {
            this.ajs = ajs;
        }
        Action.prototype.Start = function () {
            return $.ajax(this.ajs);
        };
        return Action;
    }) ();
    var HomeActions = (function () {
        function HomeActions() {}
        HomeActions.RefreshCache = function (key, returnUrl, ajaxSettings) {
            ajaxSettings = ajaxSettings || {};
            ajaxSettings['data'] = JSON.stringify({
                key: key,
                returnUrl: returnUrl
            });
            ajaxSettings['url'] = MVC.Home.ActionNames.RefreshCache;
            if ('type' in ajaxSettings == false) ajaxSettings['type'] = 'POST';
            if ('context' in ajaxSettings == false) ajaxSettings['context'] = this;
            if ('contentType' in ajaxSettings == false) ajaxSettings['contentType'] = 'application/json';
            return new Action(ajaxSettings);
        };
        HomeActions.IndexAllRecords = function (ajaxSettings) {
            ajaxSettings = ajaxSettings || {};
            ajaxSettings['data'] = JSON.stringify({
            });
            ajaxSettings['url'] = MVC.Home.ActionNames.IndexAllRecords;
            if ('type' in ajaxSettings == false) ajaxSettings['type'] = 'POST';
            if ('context' in ajaxSettings == false) ajaxSettings['context'] = this;
            if ('contentType' in ajaxSettings == false) ajaxSettings['contentType'] = 'application/json';
            return new Action(ajaxSettings);
        };
        return HomeActions;
    }) ();
    var HomeActionNames = (function () {
        function HomeActionNames() {}
        HomeActionNames.RefreshCache = '/refreshcache';
        HomeActionNames.IndexAllRecords = '/indexallrecords';
        return HomeActionNames;
    }) ();
    var Home = (function () {
        function Home() {}
        Home.Actions = HomeActions;
        Home.ActionNames = HomeActionNames;
        return Home;
    }) ();
    MVC.Home = Home;
}) (MVC || (MVC = {}));