City tax exemption is quite simple: the taxes are only due for privately arranged trips. As soon as you are traveling for business reasons, the hotel can refrain from calculating and forwarding the city tax to the municipality.
BUT: The hotels are instructed by the cities to take the tax from the room bill only if it is clearly proven that the stay is for business reasons.
Every city demands different documents, so you are exactly right here on this page. Here you will always find current information and templates that allow you to sleep tax-free in each city.