Mini Starter Swap

The big relay on the side of the mini-starter is termed "solenoid". Your original starter has one, it was just inside under the bulge in the case. In some cars (Fords I recall), the solenoid was a separate piece.

There are actually 2 relays in the starter circuit, for all cars I have owned. Your key switch actuates the "starter relay". In newer cars it is often a standard "30 A auto relay". The output of that actuates the bigger "solenoid relay", which is probably rated ~200 A.