What you describe is called a "cistron". The set of three base-pairs
within a cistron that code for a specific amino acid (or to an instruction
for the ribosome manufacturing the protein) is called a "codon". Multiple
cistrons occur on a chromosome.
The word "gene" was coined long before the discovery of DNA, and was used
to mean "whatever it is that is passed from parent to child that makes
Mendel's yellow pea plant tend to have yellow children in certain ratios,
and whatever it is that makes my bald spot and my father's no coincidence."
None of the definable units above meet that definition (except in those
rare species where recombination does not occur in mitosis, in which
case a chromosome can be called a gene).
When human sperm and egg cells undergo mitosis, there is a process called
recombination (or "crossing-over") in which the two halves of the
chromosome break in several places and swap strands of DNA. Crossover
points /do not obey cistron boundaries/, but are random. However, the
chromosome is quite long, and there are few enough crossover points,
that cistrons do tend to stay together, and in fact neighboring ones
tend to stay together in sets over many generations. It is that set of
neighboring cistrons, encoding for several proteins, and which generally
stays together enough to get passed on as a unit, that can best fit the
earlier uses of the word "gene", but it is only a statistically discrete
thing. In reality, any single mitosis might split it up, and eliminate
whatever trait its cistrons might have caused. But the individual
cistrons themselves probably wouldn't produce the same traits as the
whole set, so it is the traits produced by whole sets that compete with
their alleles in the gene pool to produce selection; individual cistrons
do not compete in this way, so they aren't "genes".
-- Lee Daniel Crocker <lee@piclab.com> <http://www.piclab.com/lcrocker.html>