Direction of surface tension

It's really both. It's like the tension in a string. You pull on one end in one direction with tension T, and someone pulls on the other end in the opposite direction with tension T.

In Fig. 1, the edge of the surface is pulling on the wall of the capillary. In Fig. 2, the wall of the capillary is pulling on the edge of the surface.

This type of behavior is encountered because, in reality, surface tension is an isotropic second order tensor defined within the surface of the liquid. In the case of a string, the tension is determined by the stress tensor (which is a second order tensor). Both the surface tension tensor and the stress tensor have this "bidirectional nature" to them.