[ResponsesAPI] Implement streaming mode

[Wauplin]

Built on top of #1576.

This PR adds support for streaming mode to the Responses API.

Tested it using the openai-responses-starter-app:

Screencast.from.02-07-2025.07.43.52.webm

pnpm run example streaming

{
  type: 'response.created',
  response: {
    object: 'response',
    id: 'resp_861131785bfb75f24f944aa7cbc4767b194a2ea320cff258',
    status: 'in_progress',
    error: null,
    instructions: null,
    model: 'Qwen/Qwen2.5-VL-7B-Instruct',
    temperature: 1,
    top_p: 1,
    created_at: 1751383702199,
    output: []
  },
  sequence_number: 0
}
{
  type: 'response.in_progress',
  response: {
    object: 'response',
    id: 'resp_861131785bfb75f24f944aa7cbc4767b194a2ea320cff258',
    status: 'in_progress',
    error: null,
    instructions: null,
    model: 'Qwen/Qwen2.5-VL-7B-Instruct',
    temperature: 1,
    top_p: 1,
    created_at: 1751383702199,
    output: []
  },
  sequence_number: 1
}
{
  type: 'response.output_item.added',
  output_index: 0,
  item: {
    id: 'msg_def4b731a2654f7eab4fb2efdff217079da37154709c0f0b',
    type: 'message',
    role: 'assistant',
    status: 'in_progress',
    content: []
  },
  sequence_number: 2
}
{
  type: 'response.content_part.added',
  item_id: 'msg_def4b731a2654f7eab4fb2efdff217079da37154709c0f0b',
  output_index: 0,
  content_index: 0,
  part: { type: 'output_text', text: '', annotations: [] },
  sequence_number: 3
}
{
  type: 'response.output_text.delta',
  item_id: 'msg_def4b731a2654f7eab4fb2efdff217079da37154709c0f0b',
  output_index: 0,
  content_index: 0,
  delta: 'Double',
  sequence_number: 4
}
{
  type: 'response.output_text.delta',
  item_id: 'msg_def4b731a2654f7eab4fb2efdff217079da37154709c0f0b',
  output_index: 0,
  content_index: 0,
  delta: ' bubble',
  sequence_number: 5
}

...

{
  type: 'response.output_text.delta',
  item_id: 'msg_def4b731a2654f7eab4fb2efdff217079da37154709c0f0b',
  output_index: 0,
  content_index: 0,
  delta: '!',
  sequence_number: 43
}
{
  type: 'response.output_text.done',
  item_id: 'msg_def4b731a2654f7eab4fb2efdff217079da37154709c0f0b',
  output_index: 0,
  content_index: 0,
  text: 'Double bubble bath! Double bubble bath! Double bubble bath! Double bubble bath! Double bubble bath! Double bubble bath! Double bubble bath! Double bubble bath! Double bubble bath! Double bubble bath!',
  sequence_number: 44
}
{
  type: 'response.content_part.done',
  item_id: 'msg_def4b731a2654f7eab4fb2efdff217079da37154709c0f0b',
  output_index: 0,
  content_index: 0,
  part: {
    type: 'output_text',
    text: 'Double bubble bath! Double bubble bath! Double bubble bath! Double bubble bath! Double bubble bath! Double bubble bath! Double bubble bath! Double bubble bath! Double bubble bath! Double bubble bath!',
    annotations: []
  },
  sequence_number: 45
}
{
  type: 'response.output_item.done',
  output_index: 0,
  item: {
    id: 'msg_def4b731a2654f7eab4fb2efdff217079da37154709c0f0b',
    type: 'message',
    role: 'assistant',
    status: 'completed',
    content: [ [Object] ]
  },
  sequence_number: 46
}
{
  type: 'response.completed',
  response: {
    object: 'response',
    id: 'resp_861131785bfb75f24f944aa7cbc4767b194a2ea320cff258',
    status: 'completed',
    error: null,
    instructions: null,
    model: 'Qwen/Qwen2.5-VL-7B-Instruct',
    temperature: 1,
    top_p: 1,
    created_at: 1751383702199,
    output: [ [Object] ]
  },
  sequence_number: 47
}

[Wauplin]

(merging so that I can continue working on new features)