Regardless of what the Shimano manual says, I would always do a gravity bleed first. Doesn’t have to be much, but you do want to flush that dirt out of the caliper and bleed nipple first rather than push it further up into the system.
This is how the Shimano techs train their service centres to do it.
1. Gravity to flush out old fluid (don’t push this back up into system again).
2. Up from caliper with brand new fluid.
3. Final gravity bleed.
Even though that’s how I’ve been shown to do it, unless they’re near brand new brakes I always do the marshy method as lubricating the pistons is the best thing you can do to help avoid wandering bite.