Firestore Repeatedly Returning Error: 5 NOT_FOUND

I found support for this [issue](https://github.com/firebase/firebase-admin-node/issues/2563) from the firebase library but the issue exists in this library as well.

cloud.google.com/support told me this was "related to code and implementation" and "falls outside the scope of GCP tech support" even though I have the _Enhanced Support_ for $500 / mo.

The issue is that if you're not using a database with the default name you have to explicitly declare the databaseID when instantiating the `Firestore` object. [These docs](https://cloud.google.com/nodejs/docs/reference/firestore/latest/firestore/firestore#examples) never mention that or display an example of a database with a custom ID. The example given:

```js
const {Firestore} = require('@google-cloud/firestore');

// Create a new client
const firestore = new Firestore();

async function quickstart() {
// Obtain a document reference.
const document = firestore.doc('posts/intro-to-firestore');

// Enter new data into the document.
await document.set({
title: 'Welcome to Firestore',
body: 'Hello World',
});
console.log('Entered new data into the document');

// Update an existing document.
await document.update({
body: 'My first Firestore app',
});
console.log('Updated an existing document');

// Read the document.
const doc = await document.get();
console.log('Read the document');

// Delete the document.
await document.delete();
console.log('Deleted the document');
}
quickstart();
```

will fail unless the custom database ID is added explicitly as so:

```js
const firestore = new Firestore({
databaseId: 'custom-name'
});
```

it's probably best if you add the `projectId` there explicitly as well:

```js
const firestore = new Firestore({
projectId: 'my-project',
databaseId: 'custom-name'
});
```

The error message for trying to write to the database without an explicit ID is unhelpful:

```bash
Error: 5 NOT_FOUND:
at callErrorFromStatus (/Users/billy/GitHub/trifle-labs/mmorps-gcp/functions/node_modules/@grpc/grpc-js/build/src/call.js:31:19)
at Object.onReceiveStatus (/Users/billy/GitHub/trifle-labs/mmorps-gcp/functions/node_modules/@grpc/grpc-js/build/src/client.js:193:76)
at Object.onReceiveStatus (/Users/billy/GitHub/trifle-labs/mmorps-gcp/functions/node_modules/@grpc/grpc-js/build/src/client-interceptors.js:360:141)
at Object.onReceiveStatus (/Users/billy/GitHub/trifle-labs/mmorps-gcp/functions/node_modules/@grpc/grpc-js/build/src/client-interceptors.js:323:181)
at /Users/billy/GitHub/trifle-labs/mmorps-gcp/functions/node_modules/@grpc/grpc-js/build/src/resolving-call.js:129:78
at process.processTicksAndRejections (node:internal/process/task_queues:77:11)
for call at
at ServiceClientImpl.makeUnaryRequest (/Users/billy/GitHub/trifle-labs/mmorps-gcp/functions/node_modules/@grpc/grpc-js/build/src/client.js:161:32)
at ServiceClientImpl. (/Users/billy/GitHub/trifle-labs/mmorps-gcp/functions/node_modules/@grpc/grpc-js/build/src/make-client.js:105:19)
at /Users/billy/GitHub/trifle-labs/mmorps-gcp/functions/node_modules/@google-cloud/firestore/build/src/v1/firestore_client.js:237:29
at /Users/billy/GitHub/trifle-labs/mmorps-gcp/functions/node_modules/google-gax/build/src/normalCalls/timeout.js:44:16
at repeat (/Users/billy/GitHub/trifle-labs/mmorps-gcp/functions/node_modules/google-gax/build/src/normalCalls/retries.js:82:25)
at /Users/billy/GitHub/trifle-labs/mmorps-gcp/functions/node_modules/google-gax/build/src/normalCalls/retries.js:125:13
at OngoingCallPromise.call (/Users/billy/GitHub/trifle-labs/mmorps-gcp/functions/node_modules/google-gax/build/src/call.js:67:27)
at NormalApiCaller.call (/Users/billy/GitHub/trifle-labs/mmorps-gcp/functions/node_modules/google-gax/build/src/normalCalls/normalApiCaller.js:34:19)
at /Users/billy/GitHub/trifle-labs/mmorps-gcp/functions/node_modules/google-gax/build/src/createApiCall.js:112:30
at process.processTicksAndRejections (node:internal/process/task_queues:95:5)
Caused by: Error
at _firestore._traceUtil.startActiveSpan (/Users/billy/GitHub/trifle-labs/mmorps-gcp/functions/node_modules/@google-cloud/firestore/build/src/write-batch.js:438:27)
at DisabledTraceUtil.startActiveSpan (/Users/billy/GitHub/trifle-labs/mmorps-gcp/functions/node_modules/@google-cloud/firestore/build/src/telemetry/disabled-trace-util.js:16:16)
at WriteBatch.commit (/Users/billy/GitHub/trifle-labs/mmorps-gcp/functions/node_modules/@google-cloud/firestore/build/src/write-batch.js:436:43)
at /Users/billy/GitHub/trifle-labs/mmorps-gcp/functions/node_modules/@google-cloud/firestore/build/src/reference/document-reference.js:354:31
at DisabledTraceUtil.startActiveSpan (/Users/billy/GitHub/trifle-labs/mmorps-gcp/functions/node_modules/@google-cloud/firestore/build/src/telemetry/disabled-trace-util.js:16:16)
at DocumentReference.set (/Users/billy/GitHub/trifle-labs/mmorps-gcp/functions/node_modules/@google-cloud/firestore/build/src/reference/document-reference.js:346:43)
at /Users/billy/GitHub/trifle-labs/mmorps-gcp/functions/index.js:35:49
at /Users/billy/GitHub/trifle-labs/mmorps-gcp/functions/node_modules/@google-cloud/functions-framework/build/src/function_wrappers.js:100:29
at process.processTicksAndRejections (node:internal/process/task_queues:77:11) {
code: 5,
details: '',
metadata: Metadata {
internalRepr: Map(1) { 'x-debug-tracking-id' => [Array] },
options: {}
},
note: 'Exception occurred in retry method that was not classified as transient'
}
```

**PLEASE READ**: If you have a support contract with Google, please create an issue in the [support console](https://cloud.google.com/support/) instead of filing on GitHub. This will ensure a timely response.

1) Is this a client library issue or a product issue?
This is the client library for "@google-cloud/firestore" .

2) Did someone already solve this?

- [x] Search the issues already opened: https://github.com/googleapis/nodejs-firestore/issues
- [x] Search the issues on our "catch-all" repository: https://github.com/googleapis/google-cloud-node
- Closest thing is [this](https://github.com/googleapis/google-cloud-node/issues/2086) which has a similarly unhelpful error message because the `projectId` was not set but no reference to `databaseId`.
- [x] Search or ask on StackOverflow (engineers monitor these tags): http://stackoverflow.com/questions/tagged/google-cloud-platform+node.js

3) Do you have a support contract?
- [x] Please create an issue in the [support console](https://cloud.google.com/support/) to ensure a timely response.

If the support paths suggested above still do not result in a resolution, please provide the following details.

#### Environment details

- OS: macOS 14.6.1 (23G93)
- Node.js version: v18.20.3
- npm version: 10.7.0
- `@google-cloud/firestore` version: 7.10.0

#### Steps to reproduce

1. Make your first database with a custom name.
2. write a doc to the collection as outlined above

Making sure to follow these steps will guarantee the quickest resolution possible.

Thanks!