Typescript Compiler Error TS2307: Cannot find module "angular2/core" when loading from CDN without NPM

I have an example of an angular2 application with typescript, but without all the node_modules in my project (like this). The necessary files are embedded in the head of my index.html (as described in example 1 here):

<script src="https://cdnjs.cloudflare.com/ajax/libs/es6-shim/0.35.0/es6-shim.min.js"></script>
<script src="https://cdnjs.cloudflare.com/ajax/libs/systemjs/0.19.16/system-polyfills.js"></script>
<script src="https://npmcdn.com/angular2/es6/dev/src/testing/shims_for_IE.js"></script>   
<script src="https://code.angularjs.org/tools/system.js"></script>
<script src="https://code.angularjs.org/tools/typescript.js"></script>
<script src="https://code.angularjs.org/2.0.0-beta.13/angular2-polyfills.js"></script>
<script src="https://code.angularjs.org/2.0.0-beta.13/Rx.js"></script>
<script src="https://code.angularjs.org/2.0.0-beta.13/angular2.dev.js"></script>

Since I want to pre-compile my typescript-files, I installed node.js and tsc. My editor (PHPStorm 2016.1) automatically compiles the files to assets/js as defined in tsconfig.json:

{
    "compilerOptions": {
        "target": "es5",
        "module": "system",
        "declaration": true,
        "noImplicitAny": false,
        "preserveConstEnums": true,
        "removeComments": true,
        "outDir": "../assets/js",
        "sourceMap": true,
        "emitDecoratorMetadata": true,
        "experimentalDecorators": true,
        "emitBOM": true,
        "moduleResolution": "node"
    }
}

The application is loaded with (as described here):

System.config({
    packages: {
        'assets/js': {
            format: 'register',
            defaultExtension: 'js'
        }
    }
});
System.import( 'assets/js/main' ).then( null, console.error.bind( console ) );

On compiling, I always get the (kind of classic) error

error TS2307: Cannot find module 'angular2/core'

PHPStorm tells me that it cannot resolve the imports:

even when all the external libraries are loaded:

But after compiling (even with the above errors) the application is working fine in the browser.

Maybe the Typescript Definition (TSD) is missing (link here), but tsd is deprecated. I read about typings install angular --ambient but I am not sure if this is really necessary (and it throws out a bunch of other errors like this).

So my question:

Is this just a side-effect of my setup (without node_modules) which I can ignore, or do I have to install other things?

My environment:

• node 5.4.1
• npm 3.3.12
• tsc 1.8.9
• Mac OS X 10.11.4
• PHPStorm 2016.1

I am a little bit lost. Any hints would be greatly appreciated.