Do I need new lugnuts?

[Frankd1]

I was installing my winter tires and steelies that I ordered from tirerack.com and I realized that the lugnuts that came with the car (RS hatchback) are for the aluminum rims so I wasn't sure if I could use them with the steels or not.

They are stepped in their design and have a washer that fits against the aluminum rim and the narrower part passes through the rim and engages the stud.

When I installed them onto the steels I noticed that even though the tips are chamfered and fit the hole in the steel wheel there is nothing for the washer to press against and they sit quite high. Should I purchase a regular set of lugnuts that are regular height and have a chamfered tip or will these be safe to install?

Any local parts store? What is the size of lugnut that I should be asking for or do I just go by the make and model of the car

Pics included:

You can see the shape of the lugnut, how they are meant to fit the OEM aluminum rim and how they fit on the aftermarket steel wheel....