Does it really matter?
Typically the President would ask for a motion to adjourn vs. making the actual motion.
However, what you need to remember is that the President is also a member and has the same right as any other member. From
Parliamentary Procedure Online FAQ page: Can the chairman make motions?
Yes, the chairman, if a member, has the same RIGHT to make a motion as any other member. In small small boards of not more than about a dozen members present, the chairman usually participates the same as other members. However, in larger assemblies, the chairman has a duty to remain impartial, so would usually not make a motion directly. The chairman could say, for example, "The chairman will entertain a motion to..." and then wait for a member to make it, or "Is there a motion to suspend the rules that interfere with hearing the speaker at this time?"
The chairman may also assume a motion, as in: "If there are no [further] corrections, the minutes stand approved as read [as corrected]." or "If there is no further business to come before the meeting, this meeting will now adjourn. [pause] Hearing none, this meeting is adjourned."
So you see, without actually directly making a motion, the chair can accomplish pretty much the same thing without blatantly compromising his or her impartiality. Another option is to ask someone before the meeting to make a motion that the chair wishes to be considered. After all, the chairman is a member, too, and has just as much right to have things go their way as any other member. Accepting the job of chairman does not remove any rights as a member. For more information, refer to RONR 10th ed. pp. 470-471, 343, 210, 234, 490.