How you ask for canter also depends on how the horse is trained. Greener horses are usually taught by bending head and neck to inside while sliding outside leg back and squeezing with calf. Doing that on more highly trained horses will cause them to throw their haunches to the inside (not be straight) during a canter depart - hence the comment about inside leg to outside rein. On more highly trained dressage horses you activate the inside leg while doing a squeeze/release on the outside rein with the outside leg sitting behind the girth preventing the hauches from coming out (think of blocking the haunches - not squeezing them).
You must also think of the riders level of expertise when teaching a rider - I certainly wouldn't tell a new rider the dressage horse cues - that's too much at the same time for someone trying to learn balance and control on horseback for the first time.
You also asked about maintaining one lead through a grid. By this I assume you mean counter canter - for example when cantering to the left horse is on left lead, when grid causes horse to canter to the right you want them to keep cantering on the left lead - which is called counter cantering. Rider has to keep outside leg back and weight on inside seatbone with slight head/neck bending to the inside (left in this case). If all goes well horse will maintain left lead canter - but rider MUST help horse mainatain balance at counter canter.