So basically if I pay e.g. 10 Sats/vByte the difference would be 880 Sats in Fees between Native & Wrapped SegWit in this example, correct?
It's a bit more complicated than that.
Weight is 3*(size with witnesses stripped) + total size (the raw byte size of all parts including witness)
Virtual size is weight / 4
This means anything that goes into witness part will increase the virtual size by 1/4x of its size while anything added to legacy part adds 1x of its size.
So when we add the 22 bytes to signature script those bytes exist both in "size with witness stripped" and "raw byte size" and when calculating weight it will be 3*22 + 22 = 4*22 ergo the virtual size is 4*22/4 = 22.
=> For 10 sat/vbyte it adds 220 satoshi.