Would "institution" be ok for ινστιτούτο? I feel like both terms are used in English with only a shade of difference in meaning (that is often ignored).
What I've found is that whereas "institute" and "institution" differ in Eng. the Gr. ινστιτούτο is used for both. So, yes both should be acceptable and it seems "institution" is the most usual meaning. So, I added it to the accepted answers. Many thanks. There is also another Gr. word: ίδρυμα. :-)