You’ve got the major difference right, but there are some others.

Eurail (no “o”) is designed for people who are not residents of Europe.  It’s good in about 18 countries (mostly Western Europe) that are members of the Eurail system.  It covers the base ticket on almost all trains in those countries (a few privately owned railroads in Switzerland don’t honor it but give a discount).  Reservations and supplements aren’t covered.

Interrail is for people who are residents of Europe. The network covers more countries than Eurail, but not all trains.  Reservations and supplements aren’t covered and it’s not honored on some of the premier “name” trains.

“eurorail”  is sometimes used (probably incorrectly) to refer to the entire rail network in Europe.

thanks oldlady[]