[codex] Add use_responses_lite 'override' logic (#26487)

## Summary

- add a defaulted `ModelInfo.use_responses_lite` catalog field
- support serializing `reasoning.context` while preserving the existing
effort and summary path
- has not been turned on for any models yet

I've added an override to parallel tools if responses_lite is on. I've
also forced persistent reasoning when using responses_lite. It would be
ideal if we could centralize all the responses_lite plumbing, but I
think this is best for now to keep the plumbing & diffs small.

## Testing

- `cargo test -p codex-protocol
model_info_defaults_availability_nux_to_none_when_omitted`
- `RUST_MIN_STACK=8388608 cargo test -p codex-core
responses_lite_sets_all_turns_context_and_disables_parallel_tool_calls`
- `RUST_MIN_STACK=8388608 cargo test -p codex-core
configured_reasoning_summary_is_sent`
- `cargo check -p codex-core --tests`
- `RUST_MIN_STACK=8388608 cargo clippy -p codex-core --tests` (passes
with pre-existing warnings in `codex-code-mode` and
`codex-core-plugins`)

codex-rs/codex-api/src/common.rs

@@ -110,12 +110,22 @@ pub enum ResponseEvent {
     ModelsEtag(String),
 }
 
+#[derive(Debug, Serialize, Clone, PartialEq)]
+#[serde(rename_all = "snake_case")]
+pub enum ReasoningContext {
+    Auto,
+    CurrentTurn,
+    AllTurns,
+}
+
 #[derive(Debug, Serialize, Clone, PartialEq)]
 pub struct Reasoning {
     #[serde(skip_serializing_if = "Option::is_none")]
     pub effort: Option<ReasoningEffortConfig>,
     #[serde(skip_serializing_if = "Option::is_none")]
     pub summary: Option<ReasoningSummaryConfig>,
+    #[serde(skip_serializing_if = "Option::is_none")]
+    pub context: Option<ReasoningContext>,
 }
 
 #[derive(Debug, Serialize, Default, Clone, PartialEq)]

codex-rs/codex-api/src/lib.rs

@@ -30,6 +30,7 @@ pub use crate::common::OpenAiVerbosity;
 pub use crate::common::RawMemory;
 pub use crate::common::RawMemoryMetadata;
 pub use crate::common::Reasoning;
+pub use crate::common::ReasoningContext;
 pub use crate::common::ResponseCreateWsRequest;
 pub use crate::common::ResponseEvent;
 pub use crate::common::ResponseStream;