As a former president and vice-president of a couple of different HOA's in the greater Phoenix area, I've had to deal with this subject both as a member of the board and as a member of the association.
You need to read the CC&R's and any other association documents you were given when you purchased your home because if the laws in your state are anything like they are here in AZ, when you purchased your home you entered into an agreement to abide by the rules of the association. If they require that you submit a request to install you must follow the requirements. In the document you referenced it states:
"The rule allows local governments, community associations and landlords to enforce restrictions that do not impair the installation, maintenance or use of the types of antennas described above, as well as restrictions needed for safety or historic preservation."
According to an association lawyer we checked with, if the association rules require that a homeowner get association approval before adding or making a change to your home, the association can require a request be submitted. Most associations won’t give you any problems with installing the dish because of the rule.
In your case, what is probably happening is the association’s management company, if you use one, is auditing the association to make sure everyone is following the rules and will ask a members to submit a request for any changes they made, something that according to the rules you should have done, so all homeowners are in compliance with the rules. It is done to protect them and you. This way they can't be accused of randomly enforcing the rules. I know first hand because they did it to me when I was on the board.
Most associations won't mess with saying no to any antenna requests that meet the limitations in your document. They are just doing it to keep all of the records up to date and everyone in compliance with the rules.
With that said, there are some associations out there that are like the Gestapo. If you have one of those you need to do what I did and get involved in the association and fight to take your association back.