Display language server info in the server logs tab (#22797)

Follow-up of https://github.com/zed-industries/zed/pull/19448

When dealing with issues like
https://github.com/zed-industries/zed/issues/22749, it's quite tedious
to ask for logs and check them out.

This PR attempts to establish a single "diagnose my language server"
place in the server logs panel, where server capabilities were already
displayed after https://github.com/zed-industries/zed/pull/19448

The design is pretty brutal, but seems to be on par with the previous
version and it's a technical corner of Zed, so seems to be ok for now:


![image](https://github.com/user-attachments/assets/3471c83a-329e-475a-8cad-af95684da960)

Release Notes:

- Improved lsp logs view to display more language server data

crates/lsp/src/lsp.rs

@@ -82,6 +82,7 @@ pub struct LanguageServer {
     outbound_tx: channel::Sender<String>,
     name: LanguageServerName,
     process_name: Arc<str>,
+    binary: LanguageServerBinary,
     capabilities: RwLock<ServerCapabilities>,
     code_action_kinds: Option<Vec<CodeActionKind>>,
     notification_handlers: Arc<Mutex<HashMap<&'static str, NotificationHandler>>>,
@@ -347,7 +348,7 @@ impl LanguageServer {
         let mut server = util::command::new_smol_command(&binary.path)
             .current_dir(working_dir)
             .args(&binary.arguments)
-            .envs(binary.env.unwrap_or_default())
+            .envs(binary.env.clone().unwrap_or_default())
             .stdin(Stdio::piped())
             .stdout(Stdio::piped())
             .stderr(Stdio::piped())
@@ -363,7 +364,7 @@ impl LanguageServer {
         let stdin = server.stdin.take().unwrap();
         let stdout = server.stdout.take().unwrap();
         let stderr = server.stderr.take().unwrap();
-        let mut server = Self::new_internal(
+        let server = Self::new_internal(
             server_id,
             server_name,
             stdin,
@@ -374,6 +375,7 @@ impl LanguageServer {
             root_path,
             working_dir,
             code_action_kinds,
+            binary,
             cx,
             move |notification| {
                 log::info!(
@@ -385,10 +387,6 @@ impl LanguageServer {
             },
         );
 
-        if let Some(name) = binary.path.file_name() {
-            server.process_name = name.to_string_lossy().into();
-        }
-
         Ok(server)
     }
 
@@ -404,6 +402,7 @@ impl LanguageServer {
         root_path: &Path,
         working_dir: &Path,
         code_action_kinds: Option<Vec<CodeActionKind>>,
+        binary: LanguageServerBinary,
         cx: AsyncAppContext,
         on_unhandled_notification: F,
     ) -> Self
@@ -466,7 +465,12 @@ impl LanguageServer {
             response_handlers,
             io_handlers,
             name: server_name,
-            process_name: Arc::default(),
+            process_name: binary
+                .path
+                .file_name()
+                .map(|name| Arc::from(name.to_string_lossy()))
+                .unwrap_or_default(),
+            binary,
             capabilities: Default::default(),
             code_action_kinds,
             next_id: Default::default(),
@@ -1055,6 +1059,11 @@ impl LanguageServer {
         &self.root_path
     }
 
+    /// Language server's binary information.
+    pub fn binary(&self) -> &LanguageServerBinary {
+        &self.binary
+    }
+
     /// Sends a RPC request to the language server.
     ///
     /// [LSP Specification](https://microsoft.github.io/language-server-protocol/specifications/lsp/3.17/specification/#requestMessage)
@@ -1278,12 +1287,13 @@ impl FakeLanguageServer {
             root,
             root,
             None,
+            binary.clone(),
             cx.clone(),
             |_| {},
         );
         server.process_name = process_name;
         let fake = FakeLanguageServer {
-            binary,
+            binary: binary.clone(),
             server: Arc::new({
                 let mut server = LanguageServer::new_internal(
                     server_id,
@@ -1296,7 +1306,8 @@ impl FakeLanguageServer {
                     root,
                     root,
                     None,
-                    cx,
+                    binary,
+                    cx.clone(),
                     move |msg| {
                         notifications_tx
                             .try_send((