Hi,
The 'Season total' precip comes from MB-defs.php $yearrn = $WX['rain0total-yearsum'];
and the value in your MBtags.php $WX['rain0total-yearsum'] = '6.25'; // rain total this year
so it is Meteobridge that is reporting that as the total rain for the year for your station.
The data value is coming from Meteobridge, and I don't see anything in Meteobridge that would allow you to set the month for when the 'Season' starts. Meteobridge may be pulling the data from your station for the 'yearly' rain.. I know that the Davis console has a month to start Yearly collection (default=January). I use October-September as a 'year' which matches our Water Year definition per the NOAA/NWS.
In short, the template is just displaying the value that comes from Meteobridge for that data. The word 'Seasonal' is used since some 'seasons' start in January (=Annual) and some don't, so 'Seasonal' was just a neutral description for 12months of data starting with the month of your choice.