Back when the Republicans ran the House, Senate and Presidency all that old stuff got rewritten. There are no legal limitations regarding how you got your appointment!
A "handbook" is not the LAW. Why should I care what a handbook says when I can look at the actual DC ST 50.1401.02(c) instead?
The law says what it says, no matter what some brain-dead DC public schools honors graduate decided to write in a handbook, and it says that reciprocity applies only to "Officers of the executive branch of the United States government who are not domiciled within the District of Columbia, whose appointment to the office held by them was by the President of the United States, subject to confirmation by the Senate, and whose tenure of office is at the pleasure of the President."
If it got rewritten years ago, then why is it still in the WestLaw District of Columbia Official Code reference which is linked from the District of Columbia official website?