To expand on that slightly, the optimizer will use the narrowest index it can to achieve whatever it needs to, because there is less data to access.
The index it chose for the 'select count(*)'  has an average of 9 bytes, whereas the clustered index is 8,345 bytes - why choose the larger index if the smaller one will achieve the same purpose?
When you use 'select *' you've asked for all of the columns and the non-clustered index can't fulfil that (at least, not alone - other work would be required) - but the clustered index can.