The US usage of "season" and the UK usage of "series" mean the same thing: a string of television episodes that are produced, and meant to air, back to back as a unit. The term derives from the fact that, in the US, broadcast television follows a seasonal pattern: broadcast shows air new episodes in the fall, winter, and spring, and take a break in the summer. While cable shows tend to vary from this pattern, the term has stuck, so we call them seasons.
In the UK, that exact same run of episodes is called a "series", because it's a single continuous "series" of episodes that are run at a time. Their shows are not necessarily as seasonally-locked as US shows are, so the "season" term isn't used that way.
For classic Doctor Who episodes, though, they were not broadcast the same way they are today. Instead of a single series, what you got was a "season's worth of serials". Each "episode" was part of a multi-part story, typically spanning between 4 and 6 episodes each. A collection of such serials (usually 8-10 of them) was a "season". While they weren't scheduled as rigidly as US seasons, there was a strong trend to air the first serial in the fall or early winter, and the last one in the following summer, that lasted from 1963 until Season 23.
The fact that the production and broadcast structure is completely different, and that everyone (including the BBC) needed a way to distinguish the 1963 "first set of episodes" from the 2005 "first set of episodes" has led to the convention that the first 7 doctors had "Seasons", 8 had "that movie", and 9+ had "series."