🎮 Give Classic, Story, Timed, and Puzzle modes a real shared lifecycle and correct rules

## Problem

The four advertised game modes do not have a coherent lifecycle and several mode-specific transformations never reach the playable state. The UI allows users to choose modes that behave like Classic or fail in edge cases.

## Evidence

- every mode exposes `initializeGame()`, but `startGame()` does not consistently use its returned word data.
- Puzzle mode creates `puzzle` metadata, while the core renderer and answer logic continue to display/spell the normal full word.
- Story mode only has story templates for a subset of categories and mutates/reads progress separately from the round lifecycle.
- Timed mode owns timer state and callbacks, but navigation, repeated starts, round completion, and mode changes do not use a shared cleanup contract.
- duplicated `loadWord()` and `showScreen()` declarations make the effective lifecycle depend on function hoisting and source order.
- mode state is persisted globally, not per child, and there are no mode tests.

## Proposed implementation

Introduce a small mode contract and a single round controller:

```text
prepareRound(context) -> round
presentWord(round, index) -> prompt
validateAttempt(prompt, answer) -> result
onAttempt(result)
onRoundEnd(reason)
dispose()
```

- Keep navigation, timers, analytics, scoring, hints, audio, and cleanup in the controller.
- Let each mode provide only its distinct prompt/validation/scoring behavior.
- Define graceful behavior for unsupported category/mode combinations.
- Make random puzzle generation deterministic under a seed and ensure anagrams are not unchanged where avoidable.

## Acceptance criteria

- Classic, Story, Timed, and Puzzle each have visibly and behaviorally distinct rules documented in the UI.
- Puzzle prompts and validation match the generated puzzle type.
- Starting, restarting, changing modes, going Home, and finishing a round leave no orphan timers/listeners/modals.
- Story mode works or is explicitly unavailable for every selectable category.
- scoring and analytics record the selected mode and its relevant result.
- mode logic is unit tested; one end-to-end test completes and exits each mode.

## Out of scope

Splitting the monolith is tracked separately in #6. This issue is about a correct behavioral contract and can be implemented alongside that refactor.