Meeting or exceeding the minimum climb gradient specified in the procedure should keep you clear of any terrain provided you remain on the procedural track. That said, any excess distance from terrain is welcomed as it allows greater safety margins (and options) in the event of a problem.
A procedure note calling for "maximum rate of climb" COULD be instigated by ATC RADAR and comm radio performance being affected by antenna site locations and associated line of sight limitations. Also the sooner an aircraft reaches the "top" of a DP the sooner it can be fitted into the enroute traffic flow.
As for how to interpret "maximum rate of climb", I would consider that to mean best AEO ROC speed at normal climb thrust/power in this case. Save the max thrust/max pitch combo for terrain and windshear escape procedures!