```python
"""Walks a courier relay directory station by station."""


def trace_relay(links: dict, start: str) -> list:
    """Walk a relay directory from start through terminal stations.

    Args:
        links: A dict mapping each station name to the name of the
            next station in the relay, with an empty string marking the
            terminal station.
        start: The name of the starting station.

    Returns:
        A list of station names visited in hand-off order, from start
        through the terminal station inclusive.

    Raises:
        ValueError: If start is not a string, is an empty string, or is
            not in the directory. If any link value is not a string,
            or if a link points to a station not in the directory, or
            if the relay contains a cycle.
    """
    if not isinstance(start, str):
        raise ValueError("start must be a string")
    if start == "":
        raise ValueError("start cannot be an empty string")
    if start not in links:
        raise ValueError("unknown starting station")

    visited: set[str] = {start}
    path: list[str] = [start]
    current: str = start

    while True:
        target = links[current]
        if not isinstance(target, str):
            raise ValueError("link value must be a string")
        if target == "":
            path.append(current)
            break
        if target not in links:
            raise ValueError("link points to a station not in the directory")
        if target in visited:
            raise ValueError("relay contains a circular chain")
        visited.add(target)
        path.append(target)
        current = target

    return path
```

This fixes both bugs:

1. **Broken links**: Now raises `ValueError("link points to a station not in the directory")` instead of silently returning early.
2. **Circular chains**: Now raises `ValueError("relay contains a circular chain")` when a station is visited twice.

It also adds the three missing validations:
- `start` must be a string (not `None`, not an int, etc.)
- `start` must not be an empty string
- Each link value must be a string (not `None`, not an int, etc.)