Corporate jet First Officer reported receiving a GPWS terrain alert on approach to LTFE; citing ATC clearance confusion as a contributing factor.

Narrative

Descending into LTFE we were high and descending quickly from an initial cruise altitude of FL430. Initially cleared on frequency 128.925 to FL250 and very quick frequency change to Approach on 124.4. Immediately cleared on check in with [the ATIS information] and QNH 1012 to descend to FL120. Then very quickly cleared to descend to 4800 ft. on QNH1012 cleared for the ILS W Runway 28R and descend with the procedure unrestricted speed.As pilot monitoring (PM) I dialed in 4800 and as transition level was 115 we were both on local altimeter setting.Pilot flying (PF) acknowledged 4800 in altitude pre select window and armed VNAV. The procedure required a crossing altitude at point FE454 of 5500 and FE453 of 4970 before crossing TEKBU at 4800 which is FAF and GS intercept. We continued down to 4800 prior to reaching point FE454 and had a subsequent terrain alert and had to climb back to 5500.There was a mixture of old and newer phraseology used. We should have been cleared to descend via the STAR. Rather than being given an altitude and then 'a descend with the procedure' clearance. Discussing it with the PF after the event he understood that the clearance to descend with the procedure was with the GS after point TEKBU the FAF.Added into this soup of confusion even though the altitudes were presented on the FMS CDU the automation did not follow the accepted VNAV profile. Unfortunately in the confusion I did not have the presence of mind to note the FMA modes that were displayed.However there is no excuse for the error. Fortunately is was a clear day and we could see we were clear of terrain however we followed the established protocol and climbed back to get on profile.