ci: template errors not being caught due. fix: error due to updated generated-types User type (#12973)

This PR consists of two separate changes. One change cannot pass CI
without the other, so both are included in this single PR.


## CI - ensure types are generated

Our website template is currently failing to build due to a type error.
This error was introduced by a change in our generated types.

Our CI did not catch this issue because it wasn't generating types /
import map before attempting to build the templates. This PR updates the
CI to generate types first.

It also updates some CI step names for improved clarity.

## Fix: type error

![Screenshot 2025-06-29 at 12 53
49@2x](https://github.com/user-attachments/assets/962f1513-bc6c-4e12-9b74-9b891c49900b)


This fixes the type error by ensuring we consistently use the _same_
generated `TypedUser` object within payload, instead of `BaseUser`.
Previously, we sometimes used the generated-types user and sometimes the
base user, which was causing type conflicts depending on what the
generated user type was.

It also deprecates the `User` type (which was essentially just
`BaseUser`), as consumers should use `TypedUser` instead. `TypedUser`
will automatically fall back to `BaseUser` if no generated types exists,
but will accept passing it a generated-types User.

Without this change, additional properties added to the user via
generated-types may cause the user object to not be accepted by
functions that only accept a `User` instead of a `TypedUser`, which is
what failed here.

## Templates: re-generate templates to update generated types

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