fix(lsp): replace @private with @nodoc for public client functions (#24415)

* fix(lsp): replace @private with @nodoc for public client functions

To prevent lua-ls warnings in plugins which use the functions.

* fix(lsp): remove duplicate type annotations/class definitions

These annotations became duplicate with https://github.com/neovim/neovim/pull/23750

runtime/lua/vim/lsp.lua

@@ -1455,7 +1455,7 @@ function lsp.start_client(config)
     end)
   end
 
-  ---@private
+  ---@nodoc
   --- Sends a request to the server.
   ---
   --- This is a thin wrapper around {client.rpc.request} with some additional
@@ -1559,7 +1559,7 @@ function lsp.start_client(config)
     return request_result
   end
 
-  ---@private
+  ---@nodoc
   --- Sends a notification to an LSP server.
   ---
   ---@param method string LSP method name.
@@ -1590,7 +1590,7 @@ function lsp.start_client(config)
     return client_active
   end
 
-  ---@private
+  ---@nodoc
   --- Cancels a request with a given request id.
   ---
   ---@param id (integer) id of request to cancel
@@ -1613,7 +1613,8 @@ function lsp.start_client(config)
   -- Track this so that we can escalate automatically if we've already tried a
   -- graceful shutdown
   local graceful_shutdown_failed = false
-  ---@private
+
+  ---@nodoc
   --- Stops a client, optionally with force.
   ---
   ---By default, it will just ask the - server to shutdown without force. If
@@ -2311,7 +2312,6 @@ function lsp.omnifunc(findstart, base)
 
   local startbyte
 
-  ---@private
   local function on_done()
     local mode = api.nvim_get_mode()['mode']
     if mode == 'i' or mode == 'ic' then