It's not explicit in the Legends book. Can you move your horse and attack in the same Combat Action? How far can you move if an attack is allowed? (Horses move 16m per combat round.)

See Charging, p. 143-144. I think your answers are all there.

For my GM, a charge is different, that's moving up to 3 times your speed and gives you a specific charging bonus.

Just to use the horse to move 8m and attack from horseback is not a listed benefit in the Mounted Combat section of the book, so he's wondering if you can only move OR attack, unless it's a charge.

My GM is a very particular fellow.

I'd say if you're just incidentally jostling around on horseback during a Combat Action, then you are free to do so in the same way you can incidentally wiggle and move on foot during combat. I would award no particular advantage to such movement, would just consider you routinely mobile with a possible passive height advantage from horseback.

If engaged, a horse would not be able to gain its full movement rate, per the rule on p. 128. The horse would be constrained at the movement increments indicated on p. 126-127.

EDIT: (as your fussy GM, I would also cap your Combat skill at your Ride ability in mounted melee, as a measure of your control of your mount)...

Oh, that's explicitly state so we've got that rule in place. Thank goodness I had read that bit in advance, so I was covered.

Horses can build momentum pretty damned fast and historically most cavalry charges were probably made at slower paces than flatout gallops.

Yes, I've generally imagined the optimal charge speed is at a canter, Gymkhana style. A flatout gallop would risk a lot of control and responsiveness.