Doors tend to open or close automatically not because their hinges are loose or slightly rotated, but because their hinge pins are not plumb. The doors could have been hung that way intentionally (or not), or their walls could have tilted as the house aged.
To have a door move on its own to the nearest open/closed position, the center of mass of the door needs to be at its high point when the door is at threshold for open/close, which in your diagram would be 60 degrees.
Imagine that the dark line at 60 degrees in your drawing is a vertical geometric plane strecthing from floor to ceiling. Both (or all 3 if that is the number) of the hinge pins must be in this plane, with the lower hinge pin 'sticking out' into the room. Thus hung, a free swinging door in a room with no significant air drafts will behave as you desire.
To accomplish this, you may need to do one or both of these:
A) shim between the hinge leaf and its mortise in the jamb
B) move the hinge screws left or right by filling the screw holes with wood splinters/dowels + glue and then re-drill pilot holes the the left or right of the previous holes.
Some ideas for shimming are:
1) A couple layers of duct tape on the back of the hinge leaf.
2) A layer or two of paperboard or non-corrugated cardboard scavenged from discarded packaging.
3) Simpson tie plates, trimmed to fit.
4) Apply spots of a thick spackle or wood filler, then install the hinge leaf, compressing the filler to achieve an exact fit. To prevent the hing from adhering to the spackle, a) wax the back of the hinge leaf with a candle, or b) use a layer of kitchen-type wax paper, or c) wrap the leaf with kitchen-type plastic wrap. When dry, remove the the hinge leaf and fill any voids with additional spackle.
Note 1: The farther 'out into the room' the lower pin, the greater the driving force.
Note 2: The heavier the door, the 'closer in' the lower pin can be to achieve the same opening/closing force.
Note 3: For uncommonly tight hinges, the door will swing freely only if the hinges are mounted such that the pins are co-linear as well. A bit of lubricant on the pins may help in this regard in intermediate cases.
Note 4: For walls/jambs that are severely out of plumb, the jambs themselves will need to be first reset to plumb.
Note 5: 1972 Patent for Center-of-Mass Hinges