For the 2,000th theory question! The leading note is often called the leading tone, albeit being one semitone below the root note. Why has it been called thus? It can't be right!
To summarize what I got out of comments above:
Tone and Note are sometimes used interchangeably, even though they probably shouldn't be.
In some parts people tend to say Tone and SemiTone (British English) where in other parts they will say Whole Step or Half Step (US English).
The phrase "leading tone" is probably not phrased in the best way since it is a half step (or semi tone) below the tonic.
To be 100% clear one can use the phrase "Leading Note". Or just realize that language is full of inconsistencies and just remember that the leading tone is just a name and may not fit the technical definition of Tone in all contexts.
OP is correct; the "leading tone" is a semitone below the tonic. In this case, "tone" referst to a note in meaning. Since the distance of a half-step is one-half of a tone, it is called a semitone. The two meanings of tone as either a note or as a specific interval are not the same, and in this case "leading tone" is referring to a note.
"Leading tone" would be a synonym of "leading note" if theorists were actually smart, but they like to make people's lives hard sometimes, so they gave the same word two meanings.