I notice in some chords the melody actually starts on a note that is not in the chord itself. For example, a song in C major, the song might start on the C chord but the melody on the D (9th). Is there a name for this? Is it the composers intention to change the C chord into a Cadd9? Is there a name for this in music?
First - chord symbols are not intended to be a full description of the music. (What you're looking for here is called 'notation').
Second, in published music they are often simplified, or even just plain wrong. We aren't seeing 'the composer's intention' - that was in the fully-notated version. We're just seeing a shorthand version someone in the publishing office prepared.
Now, consider the situation of a combo accompanying a soloist who's playing the tune (or a piano left hand playing chords under a right hand melody). What do they need to know in order to support but not clash with the tune? Obviously which basic chord, major or minor, a basic tonic shape (major 7th allowed) or dominant shape (minor 7th required). If there's a sus4, they need to know not to hit the 3rd of the chord.
Extended chords? Now we start hitting problems. The melody has a 9th. Should that note also be in the accompaniment? Probably not. So do we include it in the chord symbol? This is why I HATE accompanying a song from just the chord sheet. I need to see the melody, to know which notes to reinforce, which to avoid...
Or you may be approaching this from a jazz improvisation angle. Is that 9th in the melody integral to the chord structure or just a melodic choice? It can make the difference between an improvisation on THAT SONG or merely one over THOSE CHORDS.
So, as so often, no glib answer. Except not to confuse a chord chart with the full notation of a song.
In general, it's called a non-chord tone (as long as the melody note truly doesn't count as part of the chord). According to https://en.wikipedia.org/wiki/Nonchord_tone#Accented, there are lots of them (e.g. the overshoot-prone appoggiatura, the carried-over-from-the-previous-bar suspension, the literal accented passing tone).
Then there are the times when composers probably do intend to add chord tones. One of the most common cases I've seen in classical music of that occurring is the V13 chord.