CTRL+T does not expand or display model thinking for BYOK models

### Describe the bug

When using a BYOK (Bring Your Own Key) model with GitHub Copilot CLI, pressing CTRL+T in the terminal does not display or toggle the model's thinking/reasoning pane. Instead, nothing happens—no output, no state change, and no visible or textual indication that the model is "thinking." This occurs only with BYOK models; with standard models, CTRL+T works as expected.

### Affected version

- Copilot CLI: 1.0.48

### Steps to reproduce the behavior

1. Configure Copilot CLI to use a BYOK (Bring Your Own Key) model — e.g., **GLM5.1 from Ollama Cloud**.
2. Start a Copilot CLI session in your terminal.
3. Run a prompt or wait for the model to enter a "thinking" or reasoning state.
4. Press CTRL+T.
5. Observe that there is no indication or output from Copilot CLI and the reasoning/thinking pane does not expand or appear.

### Expected behavior

When using any supported model (including BYOK), pressing CTRL+T should expand or display the model's "thinking" or reasoning pane, showing what the model is currently considering or processing (same as with built-in models).

### BYOK configuration details

I am using the following environment variables to configure the BYOK connection (via a wrapper script at `~/.local/bin/copilot-ollama-glm`):

```bash
export COPILOT_PROVIDER_TYPE=openai
export COPILOT_PROVIDER_BASE_URL=https://ollama.com/v1/
export COPILOT_PROVIDER_WIRE_API=responses
export COPILOT_MODEL=glm-5.1:cloud
export COPILOT_PROVIDER_MAX_PROMPT_TOKENS=163840
export COPILOT_PROVIDER_MAX_OUTPUT_TOKENS=32768
export COPILOT_PROVIDER_API_KEY=
```

### Ollama Cloud API compatibility

Ollama Cloud exposes **both** OpenAI-compatible and Anthropic-compatible API endpoints:

| API | Endpoint | Purpose |
|---|---|---|
| OpenAI Chat Completions | `https://ollama.com/v1/chat/completions` | Primary API |
| OpenAI Responses | `https://ollama.com/v1/responses` | Newer Responses API (supports reasoning/thinking tokens) |
| Anthropic Messages | `https://ollama.com/v1/messages` | Secondary compatibility layer (for Claude Code integration) |

Both specs work with GLM-5.1 on Ollama Cloud. However, the **OpenAI Responses API** is the correct choice for Copilot CLI BYOK because:

1. `COPILOT_PROVIDER_WIRE_API=responses` is the wire format that carries reasoning/thinking data — this is exactly what CTRL+T toggles.
2. The Anthropic-compatible endpoint is designed for tools like Claude Code, not Copilot CLI.
3. Copilot CLI's BYOK docs explicitly support `openai` provider type with `responses` wire API.

**Key observation:** The BYOK provider is configured as `openai` provider type using the **OpenAI Responses API** wire format (`COPILOT_PROVIDER_WIRE_API=responses`), not the Anthropic Messages API. This may be relevant — the CTRL+T thinking toggle may only be wired up for native Copilot models and may not handle the OpenAI Responses API's reasoning/thinking fields for BYOK models.

**Additional note:** GLM-5.1 ignores prompts sent with `role: "developer"` — prompts must use `role: "system"` instead. The `pi-ollama-cloud` integration sets `supportsDeveloperRole: false` on all models to handle this.

### Additional context

- Terminal: Kitty 0.46.2
- OS: Linux Mint 22.3 (Zena)
- Copilot CLI version: 1.0.48
- BYOK model: GLM-5.1 from Ollama Cloud (`glm-5.1:cloud`)
- Provider: OpenAI-compatible (`COPILOT_PROVIDER_TYPE=openai`)
- Wire API: OpenAI Responses API (`COPILOT_PROVIDER_WIRE_API=responses`)
- Ollama Cloud docs: https://docs.ollama.com/api/openai-compatibility and https://docs.ollama.com/api/anthropic-compatibility
- Confirmed that CTRL+T works with non-BYOK models in the same environment.

Reproduce the issue with the reported ~/.local/bin/copilot-ollama-glm wrapper and the listed COPILOT_PROVIDER_* settings, then locate the CTRL+T handler and the BYOK OpenAI Responses reasoning path. Done means CTRL+T visibly expands or displays thinking for the reported BYOK configuration without regressing standard models.

Written by the indexing model from the issue text.