You Do Need power plug adapter while traveling from Ecuador to Saint Barthélemy.

Here is Why You Need An Adapter

The outlet and plug type used in Saint Barthélemy are C and E and in Ecuador are A and B. This is how they look like.

Ecuador

In Ecuador, outlets of type A and B are in use.

Type A

Type B

Saint Barthélemy

In Saint Barthélemy, outlets of type C and E are in use.

Type C

Type E