I would define melody as an arrangement of pitches over time which an
individual identifies as a musical "component" or "idea." The key here
is that, for a melody to exist, there must be a listener who recognizes a
melody.  This can happen internally, as when reading a score, or externally
through an actual performance.

Without a listener, and this includes composers, there is no melody;
there isn't even any music.

I remember back in school taking a music course; defining a host of
so-called musical terms constituted the final grade.  This was a very
easy course, but the foundation for the grade was garbage.  None of these
"academic" definitions ever accounted for the supreme significance of the
individual listener.  I understand the mentality.  You can't have a serious
academic subject without definitive themes and definitions; right and wrong
is crucial.  The professor of the music course was a stuck-up conducting
loser.  His primary goal in class was to appear as an artistic expert; he
did not have any recognition of the students in the class except for my
wife who infuriated him by talking to me sometimes during class.

Don Satz
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