Handle "watched phrases" API (successor to "alert words")

[gnprice]

This is a followup to:

• #663

As mentioned there, the relevant API is being expanded to support additional features:

• https://github.com/zulip/zulip/issues/28779

We should implement the new API. This is useful in particular because it makes it possible to give feedback on the API design based on the experience of implementing it.

This won't necessarily involve any change to the client's behavior, except by consuming the new API format instead of the old.