Yes, the Sun is a planet by the older definition. But the new definition isn't much better*.
The classification of what is and is not a planet changed over time. Now it is tied to some metrics involving orbits and gravity that doesn't even apply well in the Solar System let alone a different Star System.
By the original meaning the Sun and Moon (of Earth) are also planets.
There are 7 objects that visible to the naked eye (say 5.5 magnitude or less at best) which move with respect to the fixed stars (everything above 5.5 magnitude.) Sol, Mercury, Venus, Luna, Mars, Jupiter and Saturn.
The real problem is when you start using telescopes to see things moving in the "firmament" that you cannot see with the naked eye. Then you start seeing things like Vesta, Ceres, Neptune. With a powerful enough scope we can even see Uranus. All of these got called a planets at one time. Some of them are not called planets anymore. Uranus may be full of gas but still gets this top billing.
Planet was once the word for wanderer of the night sky. A nice, simple definition. Now it means something else. Somehow everyone decided once we found out there's a lot of stuff wandering out there, planet became a special status and not just a simple label. In Science and Fiction people travel to or hail from Planet X, not Oblate Spheroid Mass X in orbit around Mass Y where Mass X is much larger than anything else in a similar orbit. The most recent definitions for astronomers appears to just be trying to encode that bias giving a bad definition - complex and counter intuitive.
It's like the term spaceship. If you put a big enough engine on anything and it will fly wherever you want it. So the difference between a space station and a space ship is a question of temporary fitting.
But astronomy is full of definitions that are not well founded (asteroid belt?) Astronomy as a field needs to learn from biology. Cladistics is a big part of biological science. You're going to getting it wrong, even for 100s of years. Naming controversies come with the territory. The International Astronomer's Union is charged with naming things in space so we can all agree on what to call that thing over there. Some of names are going to be arbitrary. Turns out planet is one of those arbitrary names**.
Time is long past to just put up a list of 'these are planets' and everything else is not. Then get used to everyone else, particularly the amateurs, using different lists.
* Over long time scales even rocks are fluids. There is no such thing as a solid on the timescale of atomic stability. There are just temporary crystalline or amorphous structures seeking lower energy states. Humans (or stars) just don't live long enough (yet) to see perfect gravitational stratification of a cold bulk object by quantum tunneling.
** For fun, go look up Asimov's description of the Earth and compare with IAU's definition of a planet. Does Earth qualify?