It is a difficult question without being able to hear the arrangement by Ray Coniff. I know that the Mancini version is a Waltz..and I have heard other arrangements, all of which I recall as slow waltzs.
So, who can say? However, if you ahve downloaded several of the songs suggested on this website, and your teacher "has objections" to them...I would start to wonder. Take in something like Secret Garden, Nocturne or Run for the Roses by Kenny Loggins. These two are definitely watlzs though they are wuite different in style and tempo.
Many years ago when I first began training I worked in 3 seperate Arthur Murray studios. One was absolutely well run and had a very high standard of dance (largely due to the Dance Directors experience)...the other two were really quite poorly run and had very low standards of dance. In those studios and others I have seen a lot of "teachers" who have really only had 2 weeks of training class themselves! So, bad instructors are out there...and there are some unfortunately who would not be able to accurately choose music. It is however, almost impossible to miscount a Waltz!!
See if you can find out how long your teacher has been teaching...don't ask him/her...ask other students who have been in the studio for a long time. See what you can find out!
Best wishes, KC