Docs(examples): Add complete API programs

[tony]

Seven standalone Lua programs demonstrate connecting to a server, listing sessions/windows/panes, creating sessions/windows, querying sessions, sending input, and capturing output. Each includes imports and awaited error handling. The displayed shell launcher starts a private daemon, verifies that closing the connection leaves its bootstrap session alive, and cleans up after success or failure.

A source-owned manifest maps these complete files to eleven existing Lua API identities. The native exporter validates the targets against LuaLS declarations and includes the exact program and launcher contents. Installed source-rock checks execute the same files outside the checkout and verify their expected output and failure cleanup. Runtime APIs are unchanged; site-model integration follows separately.

Validation:

• An independent source review found no correctness blockers in the exact committed patch.
• A fresh HTTPS clone at 0395620d362ef75c74cd488f9577afda68922586 installed the documented LuaRocks dependencies with no Git URL rewrite. All seven programs passed on tmux 3.2a and 3.7c using Lua 5.5.1 and luv 1.52.1-0: fourteen positive runs and four import/runtime failure controls. All eight source file hashes matched, expected output matched, and no owned socket directories remained. Dependency setup took 68.47 seconds; positive executions totaled 10.11 seconds.
• The isolated source-rock package check passed on tmux 3.7c in 16.07 seconds, including the new examples and failure controls.
• Ten focused tooling tests passed in 0.10 seconds. Formatting passed in 0.91 seconds, lint in 3.84 seconds with no warnings, and documentation checks in 0.03 seconds. LuaLS 3.19.1 exported 114 declarations and 659 fields, resolving all eleven manifest targets and all eight complete file payloads.
• Required local mid and outer attempts both stopped at the existing five-second tooling-test subprocess deadline under host load, without reported assertion failures. Those gates remain incomplete; their budgets are unchanged. Local tooling used Python 3.12.3, which meets the documented minimum; the pinned 3.14.6 executable was unavailable.