Arrival from SKAGI… (opinion of an A330 Captain who's flown there a few times, albeit not in an A330)
Assuming that you've crossed over SKAGI at 5000 feet (MSA for N-E sector) and are at such speed that you could complete the decent to 2100 feet within the roughly 8 miles between SKAGI and D7.0IRE, and ATC has cleared you for a 'straight in approach runway...', then you would be expected to:
Leave SKAGI and decent to 2100 feet,
cross D7.0IRE,
decent at 3 degrees until D1.3IRE
decide if it's time to burn more fuel or go drink beer (Go Around or Land)
Otherwise, you'd be expected to fly the procedure as applicable.
All of that said, expect Vectors to final…
EBLG…
From NE….
Again, assuming you're at the appropriate altitude - 3300 feet - fly to LGE, then outbound on the 214 radial to either D8.5 or D12.4 as appropriate. The rest is simple.
Again, that said, expect Vectors to final.
Now, to answer your question, "…how do I know what parts of the procedure I MUST fly…". The answer will be in the form of the clearance you receive, and the level of ATC assistance available.
In your examples both airports have approach control and radar vectoring capability, therefore, the expectation is that you would likely never fly the 'full procedure' approach. However, look at the following.
In this case, the only method short of changing to some sort of visually based approach, is to fly the full procedure.

In all cases, there is always some dialog with ATC, and possibly - as is the case of Nome - with other aircraft.
That said, most often you'll hear one of the following from ATC…
"…Cleared for the ______ approach, or intercept final approach course…Clear for the approach…". Normally clearances such as these would indicate a straight in procedure is expected.
Conversely, "…cleared to the XYZ airport FOR an approach…", would indicate the use of a 'full procedure' approach, and you'd be expected to indicate which procedure you would use, and which runway you would land on.
Finally, if in doubt, ask!