Today, I looked further into the issue and finally found the solution.

Step 1.
Open the normal.dotm template. Enter the text as shown in the image below. Do not use the headings at this point. Instead, type it in and format it the way you would like the headings to appear.

Step 2.
This is actually the part that I overlooked first.

Select all your text. Click the multiple-level list button and choose the list that has heading in it. In the picture below, it's the option that is enlarged. It says kop because I'm using the Dutch version of Word 2007.

Now, you need to tell Word that you want multiple levels - not just 1 level.

Click the multiple-level list button. There's an option that lets you change the level of the list item. For each heading you want to create, change the level.

Step 3.

The last step. Select each heading and then right-click the corresponding style button in the ribbon and select "update style to selection". That may not be the exact wording (again, I'm on a Dutch copy of Word) but it should be the first item in your list. Check the image for further help.

Do this for all your headings. Save the template and press ctrl-n to start a new document. Your headings will be autonumbered.

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