USDF describes “contact” as
The reins are stretched so that they form a straight line, not a loop. “Correct contact” or “acceptance of contact” is determined by the elasticity of the connection between horse and rider. (Note: The third tier of the Training Pyramid is represented by the concept of “Connection” in the U.S., and by the concept of “Contact” [translation of “Anlehnung”] by the FEI [see Foreign Terms section].)
I believe contact and building a connection to the horse are synonymous. I like to describe contact as building a forward-feeling CONSISTENT connection through your hands to the horse’s mouth, encouraging a state within the horse’s body and psyche (when used in concert with your leg and seat) to where there is no blockage, break, or slack in the circuit that joins horse and rider into an elastic and harmonious pair.
Connection | Contact is a necessary element for Throughness.