Extract all service helpers and types from `backend-common` to the `backend-plugin-api`

[camilaibs]

Context

Ref: https://github.com/backstage/backstage/issues/24493

In preparation for the new backend system stable release, we will deprecate the backend-common package.

Description

All service helpers and types currently exported from the backend-common package should be extracted to backend-plugin-api package.

[!Note] The decision was proposed in this Spike.

Acceptance criteria

• Types and helpers should be consumable by the app and plugins without causing cyclical dependencies
• Extracted code should be re-exported from the new module in the old package
• Deprecation exports in the old packages must be declared in a deprecated.ts file
• Deprecation exports should be annotated with @deprecated and contain a comment explaining their replacements
• Other packages' import statements should be updated to use bindings from the new package
• Documentation examples should be revisited to use code from the new package instead

[camilaibs]

These are a few of the helpers to extract:

• Database: createDatabaseClient, dropDatabase, ensureDatabaseExists, isDatabaseConflictError
• Logger: coloredFormat, redactWinstonLogLine