I know, just do it and show how, but here's a new approach anyway...for the lidded box.
I have been playing with this idea, currently working on a vase.
This taller form got me thinking about the lidded box. Start with a piece of wood say 3x3x6 (maybe 7).
1) Mark the centers of the ends. (Spindle orientation markings)
2) Draw lines through the centers and up to the long side.
3) Draw a line down the center of the long sides.
4) Mark the center of the long sides.
5) Mount the blank between centers on the center marks of the long side.
6) Turn until you have a smooth curve across the ends. These will naturally have identical curves.
7) Mark the centers of the curved ends using the lines from the long sides to help.
Mount between centers. (Don't worry, the base of the box will be hollowed and the top will get a separately turned handle to cover center marks.)
9) Part the top from the base, leaving about 1/2" to 3/4" below the curve of the box top portion and with tenons on each portion for chuck mounting.
NOTE: I'd personally turn tenons for each between centers and part the two sections on the miter saw.
11) Using a compass, draw an arc on the side of the base, compass point in the mounting points, about 1/4" from the curved end.
12) Remount the bottom box section between centers on the same marks used when turning the curved ends.
13) This will be an off-center turning, with some vibration to be expected. But box size should mean not bad.
Turn the curved end down to the arc line. The difference between the lid curve and the base curve represents the thickness of the lid lip.
14) Mark center of the base and mount the base in your chuck and hollow the box, trying to keep the sides down to the lower portion of the drop lip as straight as possible. (This helps maintain a circular shape to the opening if viewed looking at the center of the opening.) Yes, probably best to shape the base a good bit before hollowing.
15) This may require 2 chucks. Finish turning the base, up to the point where it can remain in the chuck
16) Lid needs a handle which will also serve for mounting in the chuck using pen jaws or such. Probably best to drill a mortise hole on the drillpress. Base of the handle will best fit if there is a bead just above the tenon that will seat in the mortise. Turn the handle rough round, but with at least the lower half of the bead and the tenon complete.
17) Glue the handle into the top. (Epoxy may be appropriate. Would be my choice.)
18) Mount the lid in the chuck.
19) Measure the distance between the upper peaks of the box base opening and the distance between the lowest dips. Add a 1/4" to the larger distance and use it as the diameter to turn the lid to. (1/8" overlap of the lip when placed on the base.) This may be cutting it close. 1/2" instead of 1/4" may be better.
20) OK, have to take a break. Thinking of the inside curve of the lid may require some more thought...and it's lunch time. To be continued...