How do pilots manage the speed during the approach?
The Boeing chart you reference appears to use for the approach speed column the Vref speed of the aircraft at its max landing weight in the landing configuration. This is the speed you want to be at or close to when you start the landing flare (the two 747 carriers I flew for used Vref+5).

Absent charted speed limits, controller instructions, or company policy, there is no specific distance threshold along the approach path that you cannot cross above that speed. The requirement is to be at the proper speed when you enter the landing flare.

However, if you were, say, flying the classic stabilized approach, you would be in the landing configuration at this speed from the FAF (final approach fix) to the landing flare. The FAF is typically around 5 miles from the runway.

Now, insofar as speed stages, the requirement is not one of specific distance points along your approach path (again absent charted limits, etc.), but rather what you have to do speed wise while changing from a clean configuration to the landing configuration. You're approaching the airport with the gear and flaps up. The critical element speed wise on the approach is the wing configuration.

Given the 747-200 speed of 150 kts in that chart, if you slow to that speed with the flaps up, you will stall. The minimum safe speed would be around 80 kts greater than that. Go to flaps 1 (which really doesn't put the trailing edge flaps at 1 degree but rather puts out the leading edge flaps or slats as some call them). Once they're out you can safely slow to Vref+60. At flaps 5 you can slow to Vref+40. At flaps 25 you'll be able to safely slow to the Vref speed in normal operating conditions.

Other aircraft have different flap settings and different speeds. The important thing to remember is that you're changing the shape of the wing to be able to fly more slowly. Where you will be along your approach path when you do this is typically not determined by distance per se, but by time. Do it too soon and you burn more fuel. Do it too late, and the problem becomes one of not being at Vref when you need to be.