[azsdk-cli] Log to MCP clients over json-rpc instead of stdout

[benbp]

resolves #11804
fixes #13422

This PR leverages the MCP logging utility and MCP SDK logging to route output from components back to the MCP client via json-rpc. Previously we were sending back all server log messages over stderr to avoid the misleading failed to parse message log prefix from MCP clients, but it was still confusing because all logs would show up as [server stderr]. With this update logs will show up in a normal format prefixed by the category name.

Aside from visual cleanup, the intent of this change is to also give some feedback to the user when there are long-running process operations going on. The internal process helper forwards stdout by default, so with this change that output now gets sent via MCP logging. In the future we could also consider building in first class support for the MCP progress utility, but I think to start this will be sufficient.

Example mcp client logging output when running an external process:

2026-01-09 15:15:36.281 [info] Running command [pwsh -File /tmp/azsdk_example_c55940.ps1 foobar2] in /home/ben/ai/projects/mcp-progress/tools/azsdk-cli/Azure.Sdk.Tools.Cli
2026-01-09 15:15:36.287 [info] --------------------------------------------------------------------------------
2026-01-09 15:15:37.014 [info] [azsdk_example_c55940.ps1] 1: foobar2
2026-01-09 15:15:38.028 [info] [azsdk_example_c55940.ps1] 2: foobar2
2026-01-09 15:15:39.795 [error] [azsdk_example_c55940.ps1] �[31;1mWrite-Error: �[31;1mTest error message, no failure�[0m
2026-01-09 15:15:40.796 [info] [azsdk_example_c55940.ps1] 3: foobar2
2026-01-09 15:15:40.833 [info] --------------------------------------------------------------------------------
2026-01-09 15:15:40.859 [info] Server (azsdk 0.5.12.0), Client (Visual Studio Code 1.106.3) method 'tools/call' request handler completed.

[Copilot]

Pull request overview

This PR introduces MCP (Model Context Protocol) compatible logging infrastructure to the azsdk-cli tool. The changes enable proper logging when running in MCP server mode by registering custom loggers that route log messages through the MCP protocol instead of standard console output.

Key changes:

• Added new MCP logging infrastructure with context accessor, logger provider, and hosted service
• Integrated MCP context accessor into the tool instrumentation layer
• Modified service registration to conditionally use MCP-specific output helpers when in MCP mode

Reviewed changes

Copilot reviewed 6 out of 6 changed files in this pull request and generated 8 comments.

Show a summary per file

File	Description
McpLogging.cs	Core MCP logging infrastructure including context accessor, logger provider, and raw output helper
McpLoggingHostedService.cs	Hosted service to initialize and manage MCP server context lifecycle
McpLoggingExtensions.cs	Extension method to configure MCP logging services
InstrumentedTool.cs	Added MCP server context accessor injection and initialization in tool invocation
ServiceRegistrations.cs	Conditional registration of MCP raw output helper and context accessor based on output mode
Program.cs	Logging configuration changes for MCP mode with console provider filtering and MCP logging setup

[samvaity]

@benbp I think this implementation can close #13422

[benbp]

Added a couple updates:

• Removed the category name so the logger view is cleaner (users don't need to see the class FQN).
• Added a filter so process output logs don't get uploaded to azure monitor
• Refactored logger setup into its own extensions class so Program.cs is cleaner