[Doc] Describe connect across both protocol lifecycles

## Motivation and Context

The client documentation equated `MCP::Client#connect` with the MCP initialization handshake,
which MCP 2026-07-28 removed: a modern connection probes `server/discover` instead.
Calling `connect` first is still required on either era, since it settles the lifecycle
and registers the identity, capabilities, and protocol version that modern requests carry in `_meta`,
so the wording now names that rather than the handshake.

## How Has This Been Tested?

Without `connect` a modern server rejects the first request; with it the same call succeeds.
The documentation site builds with the new wording.

## Breaking Changes

None. The change is documentation prose and example comments.

Author: koic

README.md

@@ -101,7 +101,7 @@ stdio_transport = MCP::Client::Stdio.new(
 )
 client = MCP::Client.new(transport: stdio_transport)
 
-# Perform the MCP initialization handshake before sending any requests.
+# Negotiate the protocol lifecycle before sending any requests.
 client.connect
 
 # List available tools.

docs/_client/authorization.md

@@ -78,7 +78,7 @@ transport = MCP::Client::HTTP.new(
   oauth: provider,
 )
 client = MCP::Client.new(transport: transport)
-client.connect # `initialize` is sent here; if the server replies 401 the OAuth flow runs and the handshake is retried with the acquired token
+client.connect # the lifecycle is established here; if the server replies 401 the OAuth flow runs and the request is retried with the acquired token
 client.tools
 ```
 

docs/_client/lifecycle.md

@@ -12,7 +12,7 @@ This page covers `MCP::Client#connect` and how it negotiates between the two.
 
 ## Handshake
 
-Call `MCP::Client#connect` to perform the MCP [initialization handshake](https://modelcontextprotocol.io/specification/2025-11-25/basic/lifecycle#initialization) before sending any other requests. The client sends an `initialize` request through the transport, followed by the required `notifications/initialized` notification, and caches the server's `InitializeResult` (protocol version, capabilities, server info, instructions):
+Call `MCP::Client#connect` before sending any other requests. On the handshake lifecycle it performs the MCP [initialization handshake](https://modelcontextprotocol.io/specification/2025-11-25/basic/lifecycle#initialization): the client sends an `initialize` request through the transport, followed by the required `notifications/initialized` notification, and caches the server's `InitializeResult` (protocol version, capabilities, server info, instructions):
 
 ```ruby
 client.connect
@@ -22,7 +22,7 @@ client.connected?  # => true
 client.server_info # => cached InitializeResult
 ```
 
-`connect` accepts optional `client_info:`, `protocol_version:`, and `capabilities:` keyword arguments. It is idempotent: a second call returns the cached result without contacting the server. After `close`, state is cleared and `connect` will handshake again.
+`connect` accepts optional `client_info:`, `protocol_version:`, and `capabilities:` keyword arguments. It is idempotent: a second call returns the cached result without contacting the server. After `close`, state is cleared and `connect` establishes the lifecycle again.
 
 This applies to both the Stdio and HTTP transports described on the [Transports](/client/transports/) page.
 

docs/_client/transports.md

@@ -35,7 +35,7 @@ stdio_transport = MCP::Client::Stdio.new(
 )
 client = MCP::Client.new(transport: stdio_transport)
 
-# Perform the MCP initialization handshake before sending any requests.
+# Negotiate the protocol lifecycle before sending any requests.
 client.connect
 
 # List available tools.
@@ -57,7 +57,7 @@ stdio_transport.close
 The stdio transport automatically handles:
 
 - Spawning the server process with `Open3.popen3`
-- MCP protocol initialization handshake (`initialize` request + `notifications/initialized`)
+- Lifecycle negotiation (a `server/discover` probe, or `initialize` + `notifications/initialized` on the handshake lifecycle)
 - JSON-RPC 2.0 message framing over newline-delimited JSON
 - Answering server `ping` requests; see [Answering Server Pings](/client/ping/#answering-server-pings)
 
@@ -93,7 +93,7 @@ Example usage:
 http_transport = MCP::Client::HTTP.new(url: "https://api.example.com/mcp")
 client = MCP::Client.new(transport: http_transport)
 
-# Perform the MCP initialization handshake before sending any requests.
+# Negotiate the protocol lifecycle before sending any requests.
 client.connect
 
 # List available tools

docs/index.md

@@ -83,7 +83,7 @@ stdio_transport = MCP::Client::Stdio.new(
 )
 client = MCP::Client.new(transport: stdio_transport)
 
-# Perform the MCP initialization handshake before sending any requests.
+# Negotiate the protocol lifecycle before sending any requests.
 client.connect
 
 # List available tools.