How to embed SharePoint 2013 webparts directly in aspx layout page as default webparts for that layout

You can use the SPLimitedWebPart manager to add an instance of a web part at runtime. I do this on our MySites to control adding, deleting and moving web parts that the organization requires. You can put the code in the aspx page.

SPFile thePage = currentWeb.RootFolder.Files["default.aspx"]
using (Microsoft.SharePoint.WebPartPages.SPLimitedWebPartManager spLimitedWPManager = thePage.GetLimitedWebPartManager(System.Web.UI.WebControls.WebParts.PersonalizationScope.Shared))
{
       Assembly assembly = Assembly.Load("WebPartAssemblyName");
       WebPart webPart = (WebPart)assembly.CreateInstance("WebPartClassName");

       spLimitedWPManager.AddWebPart(webPart, ZoneId, ZoneIndex);
}

You may need to do something different to gain access to the Web Part Manager for your layouts page. After this you need to redirect back to the page to display the changes. You'll also want to store a bit value to ensure that you do not perform the action on each subsequent visit.

If you only need to do this once then I might recommend PowerShell instead.

Otherwise you can add the web part directly in MarkUp by registering the tag:

<%@ Register TagPrefix="ABC" Namespace="Namespace" Assembly="Assembly" %>

and directly adding the web part,

<ABC:ClassName ID="ControlID" FrameType="None" runat="server" __WebPartId="YouWebPartGUID" WebPart="true" />

but we didn't do it inside of a web zone because we did not want to allow it to be removed so I do not know if it works in that scenario. This is easiest but doesn't allow for any customization and SharePoint doesn't really "know" about the web part.

You could also deploy your page layouts as an individual feature rather than creating a whole site definition. That way you could deploy your page layouts to any SharePoint publishing site. If you are using VS 2010, start with a SharePoint Module project. Add your layout aspx file to the project. Modify the elements.xml file to resemble this:

<?xml version="1.0" encoding="utf-8"?>
<Elements xmlns="http://schemas.microsoft.com/sharepoint/">
  <Module Name="Page Layouts" Url="_catalogs/masterpage" RootWebOnly="True">
    <File Path="Page Layouts\Layout1.aspx" Url="Layout1.aspx" Type="GhostableInLibrary" IgnoreIfAlreadyExists="True">
      <Property Name="Title" Value="My Layout 1" />
      <Property Name="ContentType" Value="$Resources:cmscore,contenttype_pagelayout_name;" />
    </File>
   </Module>
</Elements>

This deploys your layout and makes it available as for new publishing pages. Now, to get a webpart to be instantiated in new pages, you modify the <File> element with the webpart definition. For example, I could define a content editor webpart to be created on new pages in Zone1 like this:

<?xml version="1.0" encoding="utf-8"?>
<Elements xmlns="http://schemas.microsoft.com/sharepoint/">
  <Module Name="Page Layouts" Url="_catalogs/masterpage" RootWebOnly="True">
    <File Path="Page Layouts\Layout1.aspx" Url="Layout1.aspx" Type="GhostableInLibrary" IgnoreIfAlreadyExists="True">
      <Property Name="Title" Value="My Layout 1" />
      <Property Name="ContentType" Value="$Resources:cmscore,contenttype_pagelayout_name;" />
      <AllUsersWebPart WebPartZoneID="Zone1" WebPartOrder="1">
        <![CDATA[ 
               <WebPart xmlns="http://schemas.microsoft.com/WebPart/v2">
                <Assembly>Microsoft.SharePoint, Version=14.0.0.0, Culture=neutral, PublicKeyToken=71e9bce111e9429c</Assembly>
                <TypeName>Microsoft.SharePoint.WebPartPages.ContentEditorWebPart</TypeName>
                <Title>Content Editor</Title>
                <FrameType>Default</FrameType>
                <FrameState>Normal</FrameState>
                <Description></Description>
                <Height />
                <Width />
                <AllowRemove>true</AllowRemove>
                <AllowZoneChange>true</AllowZoneChange>
                <AllowMinimize>true</AllowMinimize>
                <AllowConnect>true</AllowConnect>
                <AllowEdit>true</AllowEdit>
                <AllowHide>true</AllowHide>
                <IsVisible>true</IsVisible>
                <DetailLink />
                <HelpLink />
                <HelpMode>Modeless</HelpMode>
                <Dir>Default</Dir>
                <PartImageSmall />
                <MissingAssembly>Cannot import this Web Part.</MissingAssembly>
                <PartImageLarge>/_layouts/images/homepage.gif</PartImageLarge>
                <IsIncludedFilter />
                <ExportControlledProperties>true</ExportControlledProperties>
                <ContentLink xmlns="http://schemas.microsoft.com/WebPart/v2/ContentEditor" />
                <Content xmlns="http://schemas.microsoft.com/WebPart/v2/ContentEditor">
                </Content>
                <PartStorage xmlns="http://schemas.microsoft.com/WebPart/v2/ContentEditor" />
                </WebPart>        
        ]]>
      </AllUsersWebPart>
    </File>
  </Module>
</Elements>

This should be more practical than creating a whole new site definition. Hope this helps.