Ours not to reason why. I think the British Council know what they are talking about. They are some of the world's leading experts in teaching English, after all.
In most text books and reference works, 'word stress' is used to talk about the stress patterns within an individual word. (banan
ion) - as you say, apex2000, some teachers just call this underlining the stressed syllable.
'Sentence stress' is used to talk about the stress patterns within a whole sentence or phrase (I can walk
but I can't run
.'). It's a harder thing to teach than word stress, because the stressed words in a sentence can change according to what the speaker wants to say.
For teaching word stress in the classroom, I think it's worth getting students into the habit of everytime they meet a new word, recording where the stressed syllables fall. Dedicating whole lessons to the area often just depresses and confuses classes, so just do little and often.