### Describe the bug

When LiteLLM Proxy runs with **`store_model_in_db: true`** and **Redis response caching** enabled, identical `/chat/completions` requests do **not** hit the cache after the router is (re)built from Postgres deployments.

The static router init path correctly passes `cache_responses=litellm.cache is not None`, but the **DB router rebuild path** creates `litellm.Router(...)` without `cache_responses`, leaving `router.cache_responses=False` even when `litellm.cache` is initialized.

`switch_on_llm_response_caching()` runs during cache settings init, but if the router is recreated later (model sync / deployment updates), caching is not re-enabled on the new Router instance.

### Affected versions

- Reproduced on **`v1.89.4`** and **`v1.90.3`** (upstream, unpatched) - Docker image: `ghcr.io/berriai/litellm:v1.90.3`

### Configuration

```yaml general_settings: store_model_in_db: true

litellm_settings: cache: true cache_params: type: redis host: redis port: 6379 mode: default_on ttl: 3600 supported_call_types: ["completion", "acompletion"] ```

Models and credentials are stored in Postgres (not in static `model_list`).

### Steps to r…