Feature request: machine-readable (JSONL) diagnostics output

## Background

Type-checking integrations that run alongside a dev server - e.g., `vite-plugin-checker`, or a NestJS-style backend that auto-restarts once a change type-checks cleanly - need to know, after every file change, whether the project currently has type errors. There are a few ways to get that from `tsc` today, but they have downsides:

1. Re-run `tsc --noEmit` on each change. Simple to consume (process exit code), but every run is a cold check - no incremental reuse - so it's slow on a real codebase and adds latency to every save, especially on a large codebase or many changes back to back.
2. Run `tsc --watch --noEmit` once and parse its stdout. This is the fast path: the watcher reuses program state and only rechecks what changed. But the output is built for humans, not machines. This causes code like [this](https://gist.github.com/Alex-Bond/b94a5155ec06c73b9eebe7a97b3e523c). While this is faster, it's not ideal, as any changes to the response structure or copy will cause the script to break.

As a side QOL improvement, it will allow for easier logging of the errors in CI/CD - instead of trying to figure out multiline errors, it can now be easily streamed into monitoring systems.

## Proposal

A single flag that switches the entire console output stream to JSONL. This is deliberately all-or-nothing: when enabled, nothing is written as free text, so a consumer can readline → JSON.parse every line without a parser mode-switch. This also composes with --watch and --noEmit, which reuse the same reporters.

Examples of proposed output:
```jsonl
{"type":"diagnostic","category":"error","code":2304,"file":"src/a.ts","start":{"line":3,"character":9},"end":{"line":3,"character":12},"message":"Cannot find name 'foo'.","relatedInformation":[]}
{"type":"summary","errors":1,"warnings":0}
```

```jsonl
{"type":"status","code":6032,"message":"File change detected. Starting incremental compilation..."}
{"type":"diagnostic","category":"error","code":2304,"file":"src/a.ts","start":{"line":3,"character":9},"end":{"line":3,"character":12},"message":"Cannot find name 'foo'.","relatedInformation":[]}
{"type":"summary","errors":0,"warnings":0}
```

### Flag

Off the top of my head, there are a couple of ways to name the flag:

1. `--json` - while this repeats the same approach as `--pretty`, it can be confused with other configurations like `resolveJsonModule`.
2. `--pretty json` - convert `--pretty` from a simple boolean to an advanced combo of boolean and enum. The side problem is that currently, the tsconfig.json field `pretty` is a boolean, and at first glance, there are no union types in the JSON parser and validator.
3. `--diagnosticsFormat json` - the most distinct option, but can confuse when the user tries to use `--pretty` and json.

To my personal taste, it looks like microsoft/typescript-go#2 is the best option. I'm not a maintainer of this codebase, so I will leave it to core contributors to decide.

### stdout vs stderr

It might make sense to split into 2 streams for easier logging.

## Implementation

With my limited knowledge of the codebase, I can see that it is possible to implement it by creating a new flag + adding an additional formatter in diagnostics.go.

I can work on a PR for this, but I need to know the decision about the flag before I try to implement it.

Start by reading diagnostics.go and the existing handling for --pretty and --noEmit. Resolve the flag naming and stdout/stderr behavior with maintainers before implementing the additional formatter. Done means the selected mode emits machine-readable JSONL consistently, including diagnostics, summaries, and watch status messages.

Written by the indexing model from the issue text.