Centering Prayer was originally called Quiet Prayer, or Prayer of Quiet. It is in the 1917 version of the Catechism of the Catholic Church;
A degree of contemplation in which the soul experiences an extraordinary peace and rest
www.newadvent.org
The name was changed after Fr Keating and Fr Pennington gave a retreat to religious. The text of
"The Living Flame of Love" by St John of the Cross was used. In the text the word, "center," was often
used and the discussions around it at the retreat spoke of "centering prayer." As a result, the term, Quiet
Prayer, started to be called, "Centering Prayer."
St Teresa of Avila wrote of "interior prayer" often and as a Discalced Carmelite myself, it was obvious
to me that "centering prayer," and "interior prayer," were the same thing. The OCD spiritual assistant
to my OCDS croup, confirmed it when I asked him about "centering prayer."
"Into The Silent Land," by Fr Martin Laird is a good read on "contemplative prayer."