Daniels:

alphabet a type of writing system that denotes consonants and vowels

The Unicode Standard:

Alphabet. A writing system that consists of letters for the writing
of both consonants and vowels. Consonants and vowels have equal
status as letters in an alphabet. The Latin alphabet is the most
widespread and well-known example of an alphabet. The correspondence
between letters and sounds may be either more or less exact; most
alphabets do not exhibit a one-to-one correspondence between distinct
sounds (phonemes) and distinct letters (graphemes).

Daniels commented:

>(I didn't complain about this one, of course, though the first
>sentence is otiose and the last sentence applies to any writing
>system at all, and of course "grapheme" is here, as so often, used
>pretheoretically and serves no purpose.)

So, given this, one might propose the following revision:

Alphabet. A writing system in which both both consonants and vowels
are indicated. The most widespread and well-known example is the
Latin writing system. The correspondence between letters and sounds
may be either more or less exact. Many alphabets do not exhibit a
one-to-one correspondence between distinct sounds and letters or
groups of letters used to represent them; often this is an indication
of original spellings which were not changed as the language changed.

I believe that this is accurate. Comment is invited.
--
Michael Everson * * Everson Typography * * http://www.evertype.com