I read somewhere that inkjet printers have a storage compartment for
excess ink lost during printing and head cleaning and if that becomes
full then the printer won't function. Never tore one apart to see if
this was true.

I can see this reservoir. It has one side covered by a translucent plastic
film. It's nearly empty.

And the "full" indication is calculated from the number of cycles it goes
through, not a sensor (there's no sensor on the reservoir). This is one of
the counters you can see in the service menu of the firmware (as well as
total pages printed).

Can anyone help me identify the part in the photographs and their function?

Thanks,