the externalising Mind is concerned with the expression of mental forms (ideas and knowledge) in life not only by speech, but by any form it can give.

The Externalising Mind would seem to correspond in part to Cognition in Faculty Psychology, the Jungian "sensation-Thinking faculty" or "thinking primary, sensation secondary" in Jung's typology.  (Thinking Type), and - using a "fractal" approach of the Gurdjieff-Ouspensky-Bennett system - the Moving subdivision of the Intellectual subdivision of the Intellectual Center. No doubt other correlations could be found as well. According to Sri Aurobindo the Extrernalising Mind manifests through the brow (Vishuddha) chakra.

The following further quotes of Sri Aurobindo are from the "Divine Plan" website (no longer online)

People take no trouble to see whether their mind is giving them right thoughts, right conclusions, right view on things and persons, right indications about their conduct or course of action. They have their idea and accept it as truth or follow it simply because it is their idea. Even when they recognise that they have made mistakes of the mind, they do not consider it of any importance nor they try to be more careful mentally than before.

In the vital field people know that they must not follow their desires or impulses without check or control, they know that they ought to have a conscience or a moral sense which discriminates what they can or should do or what they cannot or should not do; in the field of intellect no such care is taken.

The little Mind is tied to little things:
Its sense is but the spirit's outward touch
Half-waked in a world of dark Inconscience;
It feels out for its beings and its forms
Like one left fumbling in the ignorant Night.