[Feature Request]: Use a distinct application name for GitHub Copilot chat sessions in the VS Code MSSQL extension

### Feature Description

I would like the MSSQL extension for VS Code to use a distinct and recognizable application name in the SQL Server connection string for any session initiated by GitHub Copilot chat.

In SQL Server Management Studio, queries executed through GitHub Copilot chat can be identified because they use a different application name. This makes them traceable from the DBA side and allows them to be distinguished from normal user activity.

A similar capability in the VS Code MSSQL extension would be very valuable. Queries generated or executed from Copilot, including Agent mode, should use a dedicated application name so they can be identified in:

- live activity monitoring
- Extended Events sessions
- SQL Trace or equivalent tracing setups
- DMVs such as session and request monitoring
- auditing and operational troubleshooting

Ideally, the application name should clearly indicate that the session originates from Copilot in the MSSQL extension, and it would be even better if different modes could be differentiated, for example:
- Copilot Chat
- Copilot Agent

This would improve transparency, traceability, and operational confidence when using Copilot features against SQL Server from VS Code.

### Problem and Motivation

Currently, it is difficult or impossible to reliably distinguish GitHub Copilot activity from other MSSQL extension activity in VS Code at the SQL Server level.

For DBAs, this is an important operational gap. One of the key safeguards when adopting AI-assisted tooling is the ability to observe and trace what it is doing. If Copilot-generated queries use the same application identity as normal interactive sessions, monitoring and troubleshooting become much harder.

This matters even more because Copilot is not limited to read-only assistance. In scenarios such as Agent mode, actions may result in write operations or other impactful database activity. In those cases, being able to identify that a query came from Copilot is essential for:
- auditing
- incident investigation
- performance troubleshooting
- operational confidence
- separating human-authored activity from AI-assisted activity

A distinct application name would allow DBAs to filter, trace, and review Copilot-related sessions in real time or retrospectively using standard SQL Server observability tools.

With this feature, I could monitor Copilot activity safely, create targeted traces or Extended Events sessions, and adopt the VS Code experience with much more confidence.

Expected behavior: any connection created to execute or validate Copilot chat output should set a dedicated application name that is visible from SQL Server, so Copilot activity can be monitored and traced independently from regular editor sessions.

### Related Area

- [ ] Connection dialog (SQL Server | Azure browse/Fabric browse)
- [ ] Query editor
- [ ] Query results panel
- [ ] Object Explorer
- [x] GitHub Copilot integration
- [ ] Preview/Edit data
- [ ] Table Designer
- [ ] Schema Designer
- [ ] Schema Compare
- [ ] Local SQL Server Container provisioning
- [ ] SQL database in Fabric provisioning
- [ ] DACPAC/BACPAC export/import
- [ ] SQL Database projects
- [ ] Query Plan Visualizer
- [ ] Other (please describe below)

### If you selected "Other", please describe the area

_No response_

### Confirmation

- [x] I have searched existing feature requests and couldn't find a match
- [ ] I want to help implement this feature