fix(app): only open tooltip on keyboard focus

Modal close restores focus to the tooltip's trigger, which was firing
onFocus and re-opening the tooltip. Gate onFocus on input modality so
only keyboard-driven focus opens it, matching W3C tooltip behavior.
Also removes the key-based remount workarounds that existed solely to
sidestep this bug.

packages/app/src/components/agent-status-bar.tsx

@@ -424,12 +424,7 @@ function ControlledStatusBar({
 
           {thinkingOptions && thinkingOptions.length > 0 ? (
             <>
-              <Tooltip
-                key={`thinking-${openSelector === "thinking" ? "open" : "closed"}`}
-                delayDuration={0}
-                enabledOnDesktop
-                enabledOnMobile={false}
-              >
+              <Tooltip delayDuration={0} enabledOnDesktop enabledOnMobile={false}>
                 <TooltipTrigger asChild triggerRefProp="ref">
                   <Pressable
                     ref={thinkingAnchorRef}
@@ -470,12 +465,7 @@ function ControlledStatusBar({
 
           {modeOptions && modeOptions.length > 0 ? (
             <>
-              <Tooltip
-                key={`mode-${openSelector === "mode" ? "open" : "closed"}`}
-                delayDuration={0}
-                enabledOnDesktop
-                enabledOnMobile={false}
-              >
+              <Tooltip delayDuration={0} enabledOnDesktop enabledOnMobile={false}>
                 <TooltipTrigger asChild triggerRefProp="ref">
                   <Pressable
                     ref={modeAnchorRef}

packages/app/src/components/ui/tooltip.tsx

@@ -60,6 +60,28 @@ function useTooltipContext(componentName: string): TooltipContextValue {
   return ctx;
 }
 
+// Tooltips should open on hover or keyboard focus, not when focus is restored
+// programmatically (e.g. when a Modal closes and returns focus to its opener).
+// Track the last input modality on web so TooltipTrigger can ignore focus
+// events that weren't keyboard-driven. Native has no equivalent scenario.
+let lastInputWasKeyboard = false;
+if (isWeb && typeof window !== "undefined") {
+  const markKeyboard = () => {
+    lastInputWasKeyboard = true;
+  };
+  const markPointer = () => {
+    lastInputWasKeyboard = false;
+  };
+  window.addEventListener("keydown", markKeyboard, true);
+  window.addEventListener("mousedown", markPointer, true);
+  window.addEventListener("pointerdown", markPointer, true);
+  window.addEventListener("touchstart", markPointer, true);
+}
+
+function shouldOpenOnFocus(): boolean {
+  return !isWeb || lastInputWasKeyboard;
+}
+
 function composeEventHandlers<E>(
   original?: (event: E) => void,
   injected?: (event: E) => void,
@@ -304,6 +326,7 @@ export function TooltipTrigger({
     (e: any) => {
       onFocus?.(e);
       if (!ctx.enabled || disabled) return;
+      if (!shouldOpenOnFocus()) return;
       clearOpenTimer();
       ctx.setOpen(true);
     },