才 meaning “just now”

A lot of textbooks separate this meaning of 才 from the others. In our  view though, it's actually just a subset of 才 meaning “not until”. When  used to mean “just now”, or “a moment ago”, 才 nearly always implies  that the event should have happened earlier. I.e., it's usually  saying “it didn't happen until just now”. Therefore it's just a subset  of 才 meaning “not until”, in our view.

Some examples:

我才下班。

Wǒ cái xiàbān.

I just got off work.

我才接到他的电话。

Wǒ cái jiēdàole tā de diànhuà.

I just got his call.

我才看到你的email.

Wǒ cái kàn dàole nǐ de email.

I just saw your email.

These all mean “just now” or “a moment ago”, but there's clearly a  sense of “only just now” as well. It feels as if the events were  expected to happen sooner, but didn't.

才 for emphasis

Finally, 才 can be used to add strong emphasis to a sentence. We know  that “emphasis” is often used as a default explanation for many things  when learning languages, but unfortunately in a lot of cases it is the  best explanation! 才 really is just adding heavy emphasis in these  sentences:

我才不会呢！

Wǒ cái bù huì ne!

I'd never do that!

我才不吃这么恶心的东西呢！

Wǒ cái bù chī zhème ěxīn de dōngxi ne!

There's no way I'm eating something so disgusting!

我才不要呢！

Wǒ cái bùyào ne!

Absolutely not!

你才笨呢！

Nǐ cái bèn ne!

Are you stupid or what?!

Notice how this use of 才 is combined with 呢 to really ramp up the emphasis.