Rename collaboration modes for clarity: Code → Build, Execute → Auto

I was using Claude to skim through the last few release from 0.90 and onward to get a feel for now Codex is shaping and I felt odd about the use of code and execute as mode names. Namely (Code & Execute) could more clearly communicate their distinct purposes, especially for new users encountering the modal experience for the first time.

Current state:

• Plan mode — clear, obvious purpose (thanks!).
• Code mode — ambiguous (all modes involve coding) and too generic to convey collaborative nature of the mode.
• Execute mode — feels misplaced for the intended purpose (codex engaging in more autonomous tasks following a more prescribed task/implementation plan).

Proposed renaming

• Code → Build
• Execute → Auto

Rationale

Build clearly signals collaborative implementation:

• "Let's build this feature" matches modern SWE vernacular (I'm building with Codex in real time)
• Conveys construction/crafting rather than just "writing code"
• Familiar from GitHub Projects, Linear, and other dev tools
• Distinguishes from planning (design) vs building (implementation)

Auto explicitly communicates autonomous execution:

• Clearest possible signal for "hands-off, no questions asked" (I set a plan with codex and I expect/hope for a working solution when I return)
• Shorter and punchier than "Execute"
• Matches common patterns (auto-assign, auto-merge, autopilot)
• Removes confusion with "execute" meaning "run the code"

Potential User-facing changes

TUI mode indicator would show:

• Plan mode → Build mode (cycling)
• Footer: "Build mode" instead of "Code mode"
• Plan implementation prompt: "Switch to Build and start coding" OR "switch to auto and let codex take it from here..."

API would still support both old and new names during transition for backward compatibility.

Additional context

The TUI already filters to Plan + Code only (as of #9802), hiding Pair Programming and Execute from the primary UX. This rename would make the simplified modal experience even more intuitive.

I think this is a small change with meaningful UX impact. The current Code/Execute naming works if you understand the conceptual model, but clearer names would make the modes more discoverable and self-explanatory.