In general this is correct, although there are exceptions such as "add" or "sus" chords which have different rules.a root followed by an extension number (7, 9, 11, or 13) included all previous extensions.
There is difference a between "can be omitted" and "not present." While all chord tones are implied, there are definite avoid notes that the harmonic instruments would not typically play. These not only can be, but normally should be omitted. Also the players have discretion on how to voice chords and which notes to leave out. If you have guitar and piano, it's often necessary for them to work out voicings.
Also, if the chord symbol just has the root and 7, 9, 11, or 13, it a dominant shape. The unaltered dominant chord tones and extensions include: root, major 3, perfect 5, minor 7, major 9, perfect 11, major 13.