Hi Doug,
The colour pallets that you are displaying are for different types of styling your data and will not be affected by your account type, but by the data and style type that you have selected.
Your School screenshot looks like you are using counts and amounts by size which results in a continuous colour ramp based on the theme that you select. More details about this method are available in this story map:
https://arcgis-content.maps.arcgis.com/apps/Cascade/index.html?appid=9ad60f8362a44a00ad5f52326edb1f2...
The Screen shot named free looks to be either unique symbols or predominant categories. The Smart mapping theme is selected based on your data type and chosen symbol type. If these are the same data and chosen to be styled the same, the available colours should be the same.
Check out this documentation for more information:
Change style—ArcGIS Online Help | ArcGIS
Style numbers—ArcGIS Online Help | ArcGIS
Style categories—ArcGIS Online Help | ArcGIS
Change Style reference—ArcGIS Online Help | ArcGIS
-Kelly