I've often wondered about this, as there doesn't appear to be any legislative requirement. Amongst my councils, some do and others don't and I conform to their wishes.
To add to the confusion, I note that the governance toolkit on Ask Your Council offers the following:-
"It is not normally necessary to note the name of the member moving and seconding a motion in the minutes or the fact that a vote was taken (unless a recorded vote is requested)."
"Where Standing Orders provide for the submission by notice of formal written motions to a council meeting, it is appropriate to record the names of the Councillors proposing and seconding the motion in the minutes."
Bearing in mind the fact that the model standing orders provide for the the submission by notice of formal written motions, are we advised to differentiate between the two?