Resign position but remain on board

It is usually a HOA's Bylaws that speak to (1) election of directors; and (2) appointment of officers (either president, vice-president, secretary, or treasurer). Chances are that this person who resigned as secretary is legally entitled to remain a director without any particular officer designation. In my experience with three HOAs, newly elected directors vote strictly among themselves for the officer positions, per the Bylaws. Read your Bylaws to see what they say about appointing a new secretary. It's entirely possible your Bylaws do not require the Secretary to even be a director.

Your HOA does need a secretary to record minutes. Alternatively if for some reason you cannot get anyone to be the secretary, you could ask the HOA manager to do the minutes, adding appropriate compensation. Bear in mind Minutes are supposed to be very short, generally listing only motions made and the subsequent second and any vote on the motion. The net has lots of guidance on Minutes. See especially Robert's Rules, newly revised, for guidance.

Check your documents to see what they say about resignations. As others have said, many boards select officers from among themselves, while homeowners elect, re-elect (and sometimes recall) board members. So, the secretary is correct in saying he’s still on the board although he will longer serve as secretary.

The rest of the board should pick someone else, but if no one wants it, you could hire someone to take the minutes and transcribe them, as Sue suggested, or rotate the duty among the board members who aren’t officers. For example, if you have 7 people on your board and four are officers, the remaining three should alternate taking minutes every month. No one should be recording meetings, so they must be willing and able to listen carefully and take good notes. One of these members may find the job isn’t as intimidating as they once thought and offer to take the position permanently.

One way to keep up with the proceedings is to get a copy of the agenda and prepare a document with each item listed and then you can write notes in the appropriate section, later putting them together for a formal document. That's probably what your previous secretary should have done instead of relying solely on a recording (what would he have done if the equipment failed?)

Mary

Depending on your docs, he resigned as an Officer (the Secretary) but he is still a BOD Member which is normal. Do not confuse BOD Members with Association Officers.

As to how the BOD handles the Minutes is up to them.