feat: rename product and runtime to Piik

Rename the active product, Go module, binaries, configuration, protocols and documentation to Piik while retaining the TV mascot and existing media/room behavior.

Use one current private contract without old-name aliases. Preserve SQLite authority and copy compatible local settings once; other settings start fresh. Git-message cleanup and the infrastructure name cutover are coordinated with this phase, with original history and deployment data archived externally.

Validated locally with Web and Go checks, Windows capture compilation, matched Client/Server builds, packaged Client control handshake and history-rewrite integrity checks. Skip duplicate hosted CI for this owner-authorized local release.

[skip ci]

docs/README.md

@@ -2,12 +2,12 @@
 
 [English Quick Start](../README.md) · [中文快速开始](../README.zh-CN.md)
 
-## Use Screener
+## Use Piik
 
 | What you need | Start here |
 | --- | --- |
 | Join a friend or share a screen | [Quick Start](../README.md) |
-| Run Client, choose a mode, or fix capture trouble | [Client guide](../cmd/screener-client/README.md) |
+| Run Client, choose a mode, or fix capture trouble | [Client guide](../cmd/piik-client/README.md) |
 | Collect a useful bug report | [Diagnostics and export](./reference/configuration.md#diagnostics) |
 | Host your own site | [Self-hosting](./operations/self-hosting.md) |
 | Configure ports, access or persistence | [Configuration reference](./reference/configuration.md) |
@@ -33,7 +33,7 @@ npm run dev
 In a second terminal, start the server. For a POSIX shell:
 
 ```sh
-PORT=8788 PUBLIC_BASE_URL=http://localhost:8787 go run ./cmd/screener-server
+PORT=8788 PUBLIC_BASE_URL=http://localhost:8787 go run ./cmd/piik-server
 ```
 
 Or in PowerShell:
@@ -41,8 +41,8 @@ Or in PowerShell:
 ```powershell
 $env:PORT = '8788'
 $env:PUBLIC_BASE_URL = 'http://localhost:8787'
-go build -o build/dev/screener-server.exe ./cmd/screener-server
-./build/dev/screener-server.exe
+go build -o build/dev/piik-server.exe ./cmd/piik-server
+./build/dev/piik-server.exe
 ```
 
 The fixed Windows executable path avoids repeated firewall prompts from `go run`.
@@ -87,7 +87,7 @@ Browse [all ADRs](./adr/) for their context and consequences.
   results, failed controls and limits; neither promises universal performance.
 - [Research directory](./research/): transport, capture, platform and backend evidence.
 - [Maintenance](./maintenance.md): document ownership and repository lifecycle.
-- [Licensing](../licenses/README.md): Screener's MIT scope and third-party notices.
+- [Licensing](../licenses/README.md): Piik's MIT scope and third-party notices.
 
 Research distinguishes observations from assumptions. Product modules own
 accepted behavior, source and tests own implementation detail, and Git/PRs own