[Bug] Scheduled Codex tasks lose provider binding and fail with Invalid model name

## Summary

A scheduled Codex task can fail even when the same model works in a normal Cindy conversation.

The schedule keeps the selected model (`gpt-5.6-terra`) but does not persist its provider binding. When the scheduler runs, it falls back to the default gateway, which does not expose that model.

## Reproduction

1. Create an AI automation task.
2. Select Codex with model `gpt-5.6-terra`.
3. Save the task and run it manually or wait for the schedule.

## Actual result

The run fails with:

```text
/responses: Invalid model name passed in model=gpt-5.6-terra.
Call `/v1/models` to view available models for your key.
```

## Expected result

The scheduled task should use the same configured provider as the selected model and run successfully.

## Investigation

The normal conversation record contains:

```text
model = gpt-5.6-terra
provider_id = gpt
```

But the generated scheduled task and its background session contain:

```text
model = gpt-5.6-terra
provider_id = NULL
```

As a result, the background runner uses the fallback/default gateway instead of the configured `gpt` provider.

## Environment

- Cindy 0.1.21
- Windows
- Execution mode: AI task / Codex

Start by tracing scheduled-task generation and the background session records, comparing them with the normal conversation record described in the issue. Reproduce the Codex automation with model gpt-5.6-terra and verify that the selected provider_id is persisted and used by the scheduled run without the Invalid model name error.

Written by the indexing model from the issue text.