There are a number of ways you can do this (as is usual with MODx).
1. Have a separate template for the Home page (after years of experience, this has become my preferred way of handling the common differences for the home page on most sites - while it's fun to shoehorn an entire site into one template using TVs, output modifiers and CSS, it's not really practical).
2. Use a Template Variable, with the default left empty, and just add the div and the content in the TV for the home page.
3. Put the div and content in a chunk, and use an Output Modifier in the template where you want it:
[[[[*id:is=`1`:then=`$HomeChunk`:else=``]]]]
This awkward-looking structure is best explained here
http://modx.com/blog/2012/09/14/tags-as-the-result-or-how-conditionals-are-like-mosquitoes/