You need to remove it as there is no need to download a stand alon installation. On the SQL server installation disk for 2012 it contains a feature that inclueds ssrs, ssas and ssis.
follow this instead:
http://msdn.microsoft.com/en-us/library/jj219067.aspx
EDIT
step by step guide:
http://get-sharepoint.com/2013/06/sql-2012-ssrs-integration-with-sharepoint/
EDIT
found this:
I think you mean you installed Reporting Services 2012 SP1 and chose
the SharePoint Integrated Mode.. Also, if you go to Central
Administration > General Application Settings you will notice "SQL
Server Reporting Services (2008 and 2008 R2) " this is not used to
configure Reporting Services Integration with SQL 2012 SP1...
http://social.technet.microsoft.com/Forums/en-US/bb42c91b-9195-4cf5-9b99-af48ba79d873/sharepoint-2013-and-sql-2012-sp1-reporting-services-integration-failure?forum=sharepointadmin
If you installed Reporting Services after you installed SharePoint
then you should be able to check the installation by running the
following PowerShell command by opening the SharePoint 2013 Management
Shell and pasting the command into the window. It will return the
status of SQL Server Reporting Services.
get-spserviceinstance -all |where {$_.TypeName -like "SQL Server Reporting*"}
looks like your fine ;) , just follow the steps within the first two links iv given above!
from my understanding is that sharepoint is picking up the default sql server database engine instance that is installed when you install sharepoint which is sql 2008!
EDIT
you sure your following the steps ;)
so you have sql server 2012 installed on windwos server 2012. You then went on to install ssrs feature from sql server instllation.
have you ran powerpivot for 2013 configuation?
click start -> down arrow -> under microsoft sql server 2012 click on -> reporting services configuration manager.
make sure its running!
what mode is it running under? as i belive it should be native mode?
also goto the 2008 one and disable that service. Sharepoint should pickup the correct version