Fix _isDefTermSession not propagating upon pane split/close (#13649)

## Summary of the Pull Request
In #13560 we added a member to `Pane` that lets it know if it was spawned as a default terminal session, but did not propagate that value when the pane gets split or when the pane closes. This commit fixes that. 

## Validation Steps Performed
A session spawned by a def term invocation remembers it even as it goes through splits

src/cascadia/TerminalApp/Pane.cpp

@@ -1576,11 +1576,12 @@ void Pane::_CloseChild(const bool closeFirst, const bool isDetaching)
         // Find what borders need to persist after we close the child
         _borders = _GetCommonBorders();
 
-        // take the control, profile and id of the pane that _wasn't_ closed.
+        // take the control, profile, id and isDefTermSession of the pane that _wasn't_ closed.
         _control = remainingChild->_control;
         _connectionState = remainingChild->_connectionState;
         _profile = remainingChild->_profile;
         _id = remainingChild->Id();
+        _isDefTermSession = remainingChild->_isDefTermSession;
 
         // Add our new event handler before revoking the old one.
         _connectionStateChangedToken = _control.ConnectionStateChanged({ this, &Pane::_ControlConnectionStateChangedHandler });
@@ -2472,11 +2473,12 @@ std::pair<std::shared_ptr<Pane>, std::shared_ptr<Pane>> Pane::_Split(SplitDirect
     }
     else
     {
-        //   Move our control, guid into the first one.
+        //   Move our control, guid, isDefTermSession into the first one.
         _firstChild = std::make_shared<Pane>(_profile, _control);
         _firstChild->_connectionState = std::exchange(_connectionState, ConnectionState::NotConnected);
         _profile = nullptr;
         _control = { nullptr };
+        _firstChild->_isDefTermSession = _isDefTermSession;
     }
 
     _splitState = actualSplitType;

src/cascadia/TerminalApp/Pane.h

@@ -214,21 +214,24 @@ class Pane : public std::enable_shared_from_this<Pane>
     winrt::Windows::UI::Xaml::Controls::Grid _root{};
     winrt::Windows::UI::Xaml::Controls::Border _borderFirst{};
     winrt::Windows::UI::Xaml::Controls::Border _borderSecond{};
-    winrt::Microsoft::Terminal::Control::TermControl _control{ nullptr };
-    winrt::Microsoft::Terminal::TerminalConnection::ConnectionState _connectionState{ winrt::Microsoft::Terminal::TerminalConnection::ConnectionState::NotConnected };
     static winrt::Windows::UI::Xaml::Media::SolidColorBrush s_focusedBorderBrush;
     static winrt::Windows::UI::Xaml::Media::SolidColorBrush s_unfocusedBorderBrush;
 
+#pragma region Properties that need to be transferred between child / parent panes upon splitting / closing
     std::shared_ptr<Pane> _firstChild{ nullptr };
     std::shared_ptr<Pane> _secondChild{ nullptr };
     SplitState _splitState{ SplitState::None };
     float _desiredSplitPosition;
+    winrt::Microsoft::Terminal::Control::TermControl _control{ nullptr };
+    winrt::Microsoft::Terminal::TerminalConnection::ConnectionState _connectionState{ winrt::Microsoft::Terminal::TerminalConnection::ConnectionState::NotConnected };
+    winrt::Microsoft::Terminal::Settings::Model::Profile _profile{ nullptr };
+    bool _isDefTermSession{ false };
+#pragma endregion
 
     std::optional<uint32_t> _id;
     std::weak_ptr<Pane> _parentChildPath{};
 
     bool _lastActive{ false };
-    winrt::Microsoft::Terminal::Settings::Model::Profile _profile{ nullptr };
     winrt::event_token _connectionStateChangedToken{ 0 };
     winrt::event_token _firstClosedToken{ 0 };
     winrt::event_token _secondClosedToken{ 0 };
@@ -243,8 +246,6 @@ class Pane : public std::enable_shared_from_this<Pane>
 
     bool _zoomed{ false };
 
-    bool _isDefTermSession{ false };
-
     winrt::Windows::Media::Playback::MediaPlayer _bellPlayer{ nullptr };
     winrt::Windows::Media::Playback::MediaPlayer::MediaEnded_revoker _mediaEndedRevoker;