That's the question that propelled me to start studying the suttas, and I'd say no, it's not a good interpretation.
In addition to being in the Anapanasati Sutta, all four of the aforementioned core instructional tetrads can also be found in the following canonical discourses:
-the "Greater Exhortation to Rahula Discourse" (Maha-Rahulovada Sutta, MN 62);[http://www.accesstoinsight.org/tipitaka/mn/mn.062.than.html]
-sixteen discourses of the Samyutta Nikaya's (SN) chapter 54 (Anapana-samyutta): SN 54.1, SN 54.3–SN 54.16, SN 54.20;[http://www.accesstoinsight.org /tipitaka/sn/sn54/sn54.006.than.html; http://www.accesstoinsight.org/tipitaka ... than.html; http://www.accesstoinsight.org/tipitaka ... .than.html
-the "To Girimananda Discourse" (Girimananda Sutta, AN 10.60); and,[http://www.accesstoinsight.org/tipitaka/an/an10/an10.060.piya.html] the Khuddaka Nikaya's Patisambhidamagga's section on the breath, Anapanakatha.
The first tetrad identified above (relating to bodily mindfulness) can also be found in the following discourses:
-the "Great Mindfulness Arousing Discourse" (Mahasatipatthana Sutta, DN 22)[http://www.accesstoinsight.org/tipitaka/dn/dn.22.0.than.htm] and, similarly, the "Mindfulness Arousing Discourse" (Satipatthana Sutta, MN 10),[http://www.accesstoinsight.org/tipitaka/mn/mn.010.nysa.html] in the section on Body Contemplation; and,
-the "Mindfulness concerning the Body Discourse" (Kayagatasati Sutta, MN 119) as the first type of body-centered meditation described.[http://www.accesstoinsight.org/tipitaka/mn/mn.119.than.html]