What you just described is the SPECIAL SUICA & NEX PACKAGE that only foreign passport holders can buy.Anyone can buy a regular suica card.
It is a Narita Express ticket discounted plus a Suica card with 1500 yen credit.
The Suica card can be used ANYWHERE PASMO is accepted.
Ok just to keep that simple, if you see PASMO sign SUICA is OK. 100%.
Since all private railroads and subways use PASMO thats why you don't see SUICA mentioned as much, however they are pretty much the same for the Tokyo Metropolitan area.
SUICA = PASMO
You can recharge Suica cards at PASMO machines and recharge PASMO at Suica machines. Again it's the same.