Israeli answers the question 'which country?' about a thing. It goes in front of a noun: an Israeli passport, Israeli food, the Israeli government. It has no comparative form, so nothing is 'more Israeli' in ordinary use. Keep the capital I; words built on a country name keep it in the middle of a sentence too. The word is made from the country name Israel plus a small ending, the same way Japan gives Japanese.

Here Israeli is a countable noun: one Israeli, two Israelis. It keeps the capital I. English uses the same word for the country adjective and for the person, so the article or the plural ending is what shows you which one you are reading: 'She is Israeli' describes her, 'She is an Israeli' names her as one of that group, and both are normal English.