[Hayward, 1985] defines weakly chordal graphs, also known as weakly triangulated graphs as follows:
we define a graph as weakly triangulated if neither it nor its complement contains a chordless cycle with five or more vertices
However, here, the following definition can be found:
A graph is weakly chordal if it is (anti-hole,hole)-free.
These definitions don't seem to be equivalent? A hole is a chordless cycle of length 4 or larger, whereas the first definition specifically states a chordless cycle of length 5 or larger. Am I misunderstanding this?
Finally, a Berge graph is defined as:
A Berge graph is a simple graph that contains no odd graph hole and no odd graph antihole.
The only difference with the previous definition is the 'odd' prefix. So a Berge graph is a subclass of weakly chordal graphs?
Hayward, Ryan B., Weakly triangulated graphs, J. Comb. Theory, Ser. B 39, 200-208 (1985). ZBL0551.05055.
You appear to be correct about the difference between weakly triangulated graphs and (anti-hole, hole)-free graphs. The $C_4$ shows that they are not the same. It might be the only exception though. You could consider contacting the graphclasses.org people and let them know about this (see their contact form).
About Berge graphs vs weakly triangulated graphs, I would rather say that weakly triangulated graphs form a subclass of Berge graphs (not the other way around). That is, all weakly triangulated graphs must be Berge graphs. But consider the $C_6$ graph. It is a Berge graph, but not a weakly triangulated graph.