Yes, the term "star" is actually pretty ambiguous, much like the definition of a "planet." Certainly, white dwarfs are studied by people who regard themselves as stellar astronomers, just as Pluto is studied by people who regard themselves as planetary astronomers. The most common definition of "star" requires it to have nuclear fusion going on, to distinguish stars from brown dwarfs and planets, but it's kind of silly to say that the Sun ceases to be a star when its fusion ceases, or that it only becomes a star when fusion initiates. Indeed, if you look up definitions of "star" on the web, you will get all kinds of nonsense, such as this Cambridge dictionary definition
http://dictionary.cambridge.org/us/dictionary/english/star:
"a
large ball of
burning gas in
space that is usually
seen from
earth as a
point of
light in the
sky at
night:"
That's a poor definition from the point of view of an astronomer, because stars aren't "burning" and saying what it looks like from Earth doesn't really tell us much about what it is.
Even this astronomical definition from Sky and Telescope http://www.skyandtelescope.com/astronomy-resources/what-is-a-star/ is quite poor :
"A star is a luminous ball of gas, mostly hydrogen and helium, held together by its own gravity. Nuclear fusion reactions in its core support the star against gravity and produce photons and heat, as well as small amounts of heavier elements."
That's essentially the definition of a
main-sequence star, but there are many stars that don't have fusion going on in their cores, such as when the Sun is a red giant. Many other definitions also mention fusion in the core, so are wrong by any astrophysical standard. Other definitions say that stars are spheres of gas, so that's wrong too, because rapid rotators are quite clearly not spherical. Basically, a star is any object that shares enough characteristics with the Sun, be it past, present or future, such that it is useful to group said objects into the same category as the Sun and create an astrophysical subcommunity that shares these common interests. What exactly are the properties of the Sun that make it useful to generate this subcategory of objects is really a subjective matter that is difficult to pin down, and you will be hard pressed to find a good definition as a result!