No, the A-2 visa is not limited to nationals of the sending state. The actual definition is found at 8 USC 1101(a)(15)(A)(ii):
(ii) upon a basis of reciprocity, other officials and employees [aside from "an ambassador, public minister, or career diplomatic or consular officer"] who have been accredited by a foreign government recognized de jure by the United States, who are accepted by the Secretary of State, and the members of their immediate families;
There is no mention of nationality here.
(Also, your question says that the State Department "defines several categories for A-2," but the State Department actually identifies that list as one of "examples." This implies that other examples are possible, or, to put it another way, that the list is not exclusive.)
Anyway, you shouldn't worry too much about this, because before you apply for the visa, you will have to be accredited to the US Secretary of State. I am pretty sure that the accreditation process includes mentioning your nationality. Once the State Department accepts your accreditation, you are virtually guaranteed to be approved for the visa. And the application is free of charge, of course.
Further relevant regulations are found at 22 CFR 41.21 through 41.27, concerning foreign government officials, and there is no requirement there for such officials to have the nationality of the sending state.
The Vienna Convention on Diplomatic Relations provides in Article 8 that the receiving state (here, the US) can refuse permission for members of the diplomatic staff who do not have the nationality of the sending state (here Germany). But you do not seem to be a member of the diplomatic staff, since that comprises only "the members of the staff of the mission having diplomatic rank." If you were, you would be receiving an A-1 visa (as "an ambassador, public minister, or career diplomatic or consular officer").
In the terms of the convention, therefore, you seem to be a member of the administrative, technical, or service staff of the mission, and there is nothing to be found in the convention concerning limitations on nationality. Even if there were, the US could still choose to accredit you, of course.
Don't forget to go to the diplomats' line when you arrive; it's generally much faster than everything else, and you won't have to remind the immigration inspector not to take your fingerprints (the line is usually labeled for holders of A and G visas, or A, G, and NATO visas; the last time I saw one, it was also accepting active-duty US military traveling on official orders).