Splitting raster by polygons and defining output names using ArcMap

The Split Raster tool allows you to only specify a prefix that will be applied to each of the raster datasets you will create. A number then will be appended to each prefix, starting with 0.

If you want to have control over the names of the rasters, instead of using Split Raster you can make a loop that uses the Extract by Mask tool to clip the raster by one polygon at a time by selecting that polygon before executing the tool.

Here is the documentation of the tool: https://desktop.arcgis.com/en/arcmap/10.3/tools/spatial-analyst-toolbox/extract-by-mask.htm

A fairly simple model will achieve this. Iterate over your featureclass using the extract by mask to "cookie cut" out your raster.

You need to be using inline variable substitution to pass the ID to the output raster name, don't know what that is then read the help file!

Also you'll need to set the snap raster environment setting to ensure consistent alignment and match the cell size of your DEM. The temporary copy of your catchment polygon is used to set the extent environment setting of the extract by mask.

Below is the model that will achieve what you want.