Applying a rich black is a good idea for going to professional printers. But it's rarely a good idea when printing from a home printer/digital printer.
I find it best to go the Edit> Preferences and in the Appearance of Black change the output to Rich Black.
You can then leave the Black as 100% black in the layout - and output to an RGB device as Rich Black.
NOTE: This only affects RGB devices (regardless of whether it uses CMYK or RGB inks the device is inheritently RGB)
In regards to Photoshop - it's not the same as Indesign, that's why it doesn't behave the same.