chore: rename ChatGpt -> Chatgpt in type names (#10244)

When using ChatGPT in names of types, we should be consistent, so this
renames some types with `ChatGpt` in the name to `Chatgpt`. From
https://rust-lang.github.io/api-guidelines/naming.html:

> In `UpperCamelCase`, acronyms and contractions of compound words count
as one word: use `Uuid` rather than `UUID`, `Usize` rather than `USize`
or `Stdin` rather than `StdIn`. In `snake_case`, acronyms and
contractions are lower-cased: `is_xid_start`.

This PR updates existing uses of `ChatGpt` and changes them to
`Chatgpt`. Though in all cases where it could affect the wire format, I
visually inspected that we don't change anything there. That said, this
_will_ change the codegen because it will affect the spelling of type
names.

For example, this renames `AuthMode::ChatGPT` to `AuthMode::Chatgpt` in
`app-server-protocol`, but the wire format is still `"chatgpt"`.

This PR also updates a number of types in `codex-rs/core/src/auth.rs`.

codex-rs/app-server/src/codex_message_processor.rs

@@ -1246,7 +1246,7 @@ impl CodexMessageProcessor {
         let account = match self.auth_manager.auth_cached() {
             Some(auth) => Some(match auth {
                 CodexAuth::ApiKey(_) => Account::ApiKey {},
-                CodexAuth::ChatGpt(_) | CodexAuth::ChatGptAuthTokens(_) => {
+                CodexAuth::Chatgpt(_) | CodexAuth::ChatgptAuthTokens(_) => {
                     let email = auth.get_account_email();
                     let plan_type = auth.account_plan_type();
 

codex-rs/app-server/tests/common/auth_fixtures.rs

@@ -159,7 +159,7 @@ pub fn write_chatgpt_auth(
     let last_refresh = fixture.last_refresh.unwrap_or_else(|| Some(Utc::now()));
 
     let auth = AuthDotJson {
-        auth_mode: Some(AuthMode::ChatGPT),
+        auth_mode: Some(AuthMode::Chatgpt),
         openai_api_key: None,
         tokens: Some(tokens),
         last_refresh,