I think it would help your students understand by clarifying the following three different terms.
Chord - A chord is a combination of 3 or more pitches played simultaneously. If a note is played, it is part of the chord; if it is not played, it is not part of the chord. This may be what your student is referring to.
Harmony - A harmony is a theoretical construct that offers context to a chord and insight into its function within a piece. A C major harmony, for example, contains all Cs, all Es, and all Gs at every octave. A chord is typically a manifestation of a harmony containing a few of its notes; without a chord, you cannot play a harmony directly.
Fingering - A fingering is, of course, an instruction on what to do with your left hand, typically involving pressing or damping some or all of the strings. A fingering will usually conform to the harmony, but in a range constrained by the physical limitations of the instrument and the musician's hands. Not all notes in a fingering need to be sounded; for example, a musician might finger a C major open chord but not sound the low string-- in fact, they might even use their pinky to damp the low string so they can easily play the harmony in root position.
All three of these terms describe very similar things that are nonetheless different in important ways.
So your student is right; technically, the chord is what you play, and the harmony and fingering are something different.