Let’s define these visually.
This is an arbor (an outdoor archway):
This arbor was built by Handyman Project Manager Chris Ryan for a customer.
This is an arbor with a trellis (latticework to support climbing plants):
The pictured arbor was built by Project Manager Randy Boyer for his garden. The top of this arbor is very much like pergola…
There are also pergolas which are posts with a simple roof structure:
This pergola was built by Foreman Duane Overlock for a customer.
This is the entrance of the customer’s home which has a mock pergola: