In Your meeting, Mr. A is present and Mr. C is special invitee.
I will suggest you, to firstly introduce Mr. C as additional director and then you resign Mr. B.
As per your queries, appointment of director where director of the company is less than the quorum fixed by article for the meeting of the Board. In your case the, the quorum is not, because of the resignation of one director out of the two director.
In such case you must follow the provision of your company articles read with clause 75 of the Schedule I- Table A of the Companies Act, 1956.
Mr. A will sign the Minutes or the Chairman of the next Board meeting will sign the minutes of this Board Meeting.
Generally in corporate practice Mr. A is eligible to sign the minutes, because he is appointed as chairman for the meeting. You have also option to sign the minuted by Mr. C if he is appointed as Chairman of next meeting and then he will sign the minutes of last meeting.
In Your case Mr. B, is not authorise to sign minutes, in any case.