Telit cellular module pins connection
No where in the text you copied does it say that HW_SHUTDOWN* is internally pulled up. Instead it states that this pin must not be left floating. Since it is active low for shutdown, I’d tie it to Vcc (1.8V).

As for ON_OFF*, while it is internally pulled up, you must pulse it low for at least 5 seconds to activate the unit, as shown in section 5.2 that you included.

The text you included seems quite clear. I’m not sure what is confusing you.