If it's the Apple-brand cable, I'm 99% sure it'll work just fine - I just grabbed an old cable that came with a 2016 adapter, threw it on the SuperPort and iStat Menus is reporting 100 watts - and it charged the 16" very quickly.
Your links are apple.com, so they're fine - but beware of nearly identical looking Amazon cables - most are NOT 100W. Many are 30W or less (meant for charging phones, tablets and small laptops, not 16" workstations). The only two cables I know for sure (from testing) are 100W are the Apple cables from 15" and 16" MacBook Pros (any year) and the Zendure cable that comes with the SuperPort.
Some of the OWC Thunderbolt 3 cables claim 100W, and I've never known OWC to mis-specify anything, so I'm quite sure they really are. There are probably others (some Dell cables should be, for example), but an average USB-C cable is NOT.
I'm pretty sure that what ships with the adapter is NOT a Thunderbolt 3 cable (I've never actually tried using an adapter cable as a data cable, but a long Thunderbolt 3 cable is expensive enough that I'd be very surprised if Apple is throwing them in attached to adapters). A 2 meter OWC Thunderbolt 3 cable is $34.99 for a half-speed 20 Gb/S cable and $64.99 for a 40 Gb/S cable.
Note that the usual Apple charger cable is 2 meters, and both cables that you have linked to are much shorter. If you're using an adapter that plugs directly into the wall, a short cable will be a pain!