The simple message of the books and Sage Advice is, "you only take damage once per turn". From the SA on moonbeam, "You can’t move a creature in and out of it to damage it over and over again on the same turn."
If someone uses thunderwave to move your character into the area of insect plague, then you take damage for entering the area. You are taking this damage on their turn. The spell says "…enters the spell’s area for the first time on a turn…", not "on your turn".
When it is your turn, if you don't leave the area, then you take damage at the end of your turn (the second time you've taken damage, but the first time on this turn).
If on your turn, you leave the area then enter it again, you take damage a second time, for entering the area on your turn. Again, this is the first time you've taken the damage on this turn.
If you leave the area, then enter it (taking damage), then end your turn in the area, you do not take damage for ending your turn in the area of effect. You've already taken the damage once this turn.
I believe that this rule is for the sake of not being mean. Imaging someone grappling a character and moving them in and out and in and out, damaging them each time. It also makes the game run quicker, by not rolling dice multiple times each turn.