[atarutin]

Problem

GenContext has one SchemaGenerator (generate.rs, schema field), built with
schemars' default contract, Deserialize. OperationOutput impls such as axum::Json<T>
(axum/outputs.rs) use that same generator, so response schemas say what serde accepts,
not what it produces. Example: Option<T> without skip_serializing_if is always present
in a response, but the response schema marks it optional. TypeScript generated from the
document then has field?: T | null where the right type is field: T | null.

#254 tried two generators and stalled on the question raised in review: what happens when a
type appears on both sides? Both generators emit #/components/schemas/Foo, and a plain
extend lets one silently overwrite the other.

Fix

• GenContext gets a second generator. schema (inputs) is now explicitly
  for_deserialize(). The new output_schema is for_serialize(). axum::Json and
  axum::Form outputs use output_schema, with matching resolve_output_schema and
  apply_output_transforms methods.
• The output generator writes its $refs under a private path
  (#/components/x-aide-output-schemas/), so input and output references can always be
  told apart, including references nested inside other components.
• When the document is finished (ApiRouter::finish_api*), take_merged_definitions merges
  the two sets of definitions:
  • A name is split if its schemas differ, or if either schema refers to a split name.
    This runs to a fixed point, because Outer { inner: Rating } has the same text under
    both contracts but means different things once Rating is split.
  • Names that are not split become one shared component under their usual name.
  • Split names become {Name}-Input and {Name}-Output. FastAPI/Pydantic do the same with
    separate_input_output_schemas.
  • Every $ref in paths, webhooks and components is rewritten to match. If a name
    still collides, generation panics with a clear message instead of overwriting silently.

The result does not depend on route registration order, because both generators are kept
until the end.

Breaking for 0.16: response components gain required entries, and types used on both
sides with differing schemas get new component names. Users who replace ctx.schema with
custom settings should now also set ctx.output_schema.

Tests

crates/aide/src/axum/contract_tests.rs (feature axum-json + axum-query):

• response_option_without_skip_is_required
• response_option_with_skip_stays_optional
• request_option_stays_optional
• query_option_stays_optional
• type_on_both_sides_is_split_and_refs_follow_the_side: covers the split, the transitive
  split, and a shared component that is not split
• every_ref_resolves_and_no_private_path_leaks

Verification

• On main, with only the test file added, 2 of the 6 tests fail
  (response_option_without_skip_is_required, type_on_both_sides_is_split...).
• On this branch:
  • cargo test at the workspace root (CI's command) passes 19 lib and 18 doc tests.
  • cargo test -p aide --all-features passes 27 lib and 18 doc tests.
  • cargo fmt --check is clean.
  • cargo clippy -p aide --all-features --all-targets gives the same 9 warnings as main.
• cargo test -p aide with default features does not compile on main either, because the
  operation.rs tests need macros.

Fixes #250. This builds on the two-generator idea in #254 and answers the review question
there about types used on both sides.