eas update --channel ineffective

### Build/Submit details page URL

_No response_

### Summary

Can't set the `channel` variables when I run `eas update --channel staging`, but it works for `eas update`.

### Managed or bare?

Managed

### Environment

```
expo-env-info 1.0.5 environment info:
System:
OS: Windows 10 10.0.22621
Binaries:
Node: 18.16.0 - C:\Program Files\nodejs\node.EXE
Yarn: 1.22.5 - C:\Program Files (x86)\Yarn\bin\yarn.CMD
npm: 9.6.6 - C:\Program Files\nodejs\npm.CMD
IDEs:
Android Studio: AI-222.4459.24.2221.9862592
npmPackages:
babel-preset-expo: ^9.3.0 => 9.5.0
expo: ^48.0.19 => 48.0.20
react: 18.2.0 => 18.2.0
react-native: 0.71.8 => 0.71.8
Expo Workflow: managed
```

```
✔ Check Expo config for common issues
✔ Check package.json for common issues
✔ Check dependencies for packages that should not be installed directly
✔ Check for common project setup issues
✔ Check npm/ yarn versions
✔ Check Expo config (app.json/ app.config.js) schema
✔ Check that native modules do not use incompatible support packages
✔ Check for legacy global CLI installed locally
✖ Check that native modules use compatible support package versions for installed Expo SDK
✖ Check that packages match versions required by installed Expo SDK

Detailed check results:

Expected package @expo/config-plugins@~6.0.0
Found invalid:
@expo/config-plugins@4.1.5
(for more info, run: npm why @expo/config-plugins)
Advice: Upgrade dependencies that are using the invalid package versions.

Some dependencies are incompatible with the installed expo version:
sentry-expo@4.2.0 - expected version: ~6.2.0
Your project may not work correctly until you install the correct versions of the packages.
Install individual packages by running npx expo install sentry-expo@~6.2.0
Found outdated dependencies
Advice: Use 'npx expo install --check' to review and upgrade your dependencies.

One or more checks failed, indicating possible issues with the project.
```

### Error output

_No response_

### Reproducible demo or steps to reproduce from a blank project

I have an **eas.json** that looks like this:
```
{
"cli": {
"version": ">= 3.16.0"
},
"build": {
"android": {
"autoIncrement": true
},
"ios": {
"autoIncrement": true
},
"development": {
"channel": "development",
"developmentClient": true,
"distribution": "internal",
"env": {
"EXPO_PUBLIC_ENV": "development"
}
},
"preview": {
"distribution": "internal"
},
"production": {
"channel": "production",
"android": {
"buildType": "apk"
},
"env": {
"EXPO_PUBLIC_ENV": "production"
}
},
"staging": {
"channel": "staging",
"android": {
"buildType": "apk"
},
"env": {
"EXPO_PUBLIC_ENV": "staging"
}
}
},
"submit": {
"production": {}
}
}
```

When I run `eas build --profile staging` everything works great, I have a page where I print `Updates.channel` and the result is `staging`.

When I run `eas update --channel staging` and then open the app, the page shows that `Updates.channel` is empty.

QUESTION: how can I run `eas update` and still have the same `Updates.channel = "staging"`?