chore: backport PR #20775 to v4-next (auto-enqueue public init nullifier)

[AztecBot]

Summary

Backport of #20775 to v4-next.

feat!: auto-enqueue public init nullifier for contracts with public functions

Contract initialization now emits two separate nullifiers (private + public) instead of one, fixing a bug where public functions could be called on an uninitialized contract if enqueued before the initializer's public calls.

Backport details

Cherry-picked merge commit cee97a1ba434f8469245152ca18d306f33d303fb onto latest backport-to-v4-next-staging.

Conflicts resolved in:

• docs/docs-developers/docs/resources/migration_notes.md — added migration notes for the new init nullifier behavior
• noir-projects/aztec-nr/aztec/src/macros/aztec.nr — adapted imports for v4-next style, added emit_public_init_nullifier import and quote output alongside existing offchain_receive
• noir-projects/noir-protocol-circuits/crates/types/src/constants.nr — added DOM_SEP__PUBLIC_INITIALIZATION_NULLIFIER
• noir-projects/noir-protocol-circuits/crates/types/src/constants_tests.nr — adjusted test counts from <50, 43> to <51, 44>
• yarn-project/end-to-end/src/composed/ha/e2e_ha_full.test.ts — excluded reloadKeystore test (context from next, not part of this PR)

Verification

All modified Noir contracts (aztec-nr, init_test_contract, private_init_test_contract, stateful_test_contract) compile successfully with nargo check.

Supersedes #21419 (based on older v4-next).

ClaudeBox log: https://claudebox.work/s/ff2dd4f68e9ac0be?run=2