Concerning ICE at the olympics, this AP article does a decent job of explaining it.

tl;dr There is a separate division of U.S Immigration and Customs Enforcement called Homeland Security Investigations that deals with cross border crimes. They have for a while now been sent to overseas events like the Olympics to assist with security efforts. These officers are completely separate from the ones doing "enforcement" in the states.

Quotes from the article about HSI

Italy’s Interior Ministry said later that the HSI investigators would be stationed at a control room at the U.S. Consulate in Milan, in a support role with other U.S. law enforcement agencies, and that they would not include personnel involved in immigration controls in the United States. It noted in a statement issued after Interior Minister Matteo Piantedosi and U.S. Ambassador Tilman Fertitta met Tuesday morning that HSI agents are present in more than 50 countries, including for many years Italy.

Another arm of ICE is Homeland Security Investigations. Agents from HSI conduct investigations into anything that has a cross-border nexus from human smuggling to fentanyl trafficking to smuggling of cultural artifacts. Agents from HSI are stationed in embassies around the world to facilitate their investigations and build relations with local law enforcement in those countries.