chilenisch
Chilean
Relating to the country of Chile, its people, or its culture.
Wir haben gestern einen ausgezeichneten chilenischen Wein getrunken.
Yesterday we drank an excellent Chilean wine.

Chilean
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Chilean
Relating to the country of Chile, its people, or its culture.
Wir haben gestern einen ausgezeichneten chilenischen Wein getrunken.
Yesterday we drank an excellent Chilean wine.

The adjective is derived from the country name 'Chile' with the ending '-isch'. This pattern is used for many adjectives that describe origin, e.g. 'italienisch' (Italian), 'japanisch' (Japanese).
Die chilenische Flagge hat die Farben Weiß, Blau und Rot.
The Chilean flag has the colors white, blue, and red.
Adjectives derived from geographical names and ending in '-isch' are always written in lowercase in German. So it is 'der chilenische Wein' (the Chilean wine), not 'der Chilenische Wein'.
Ich lese gerade einen Roman einer chilenischen Autorin.
I am currently reading a novel by a Chilean author.
'Chilenisch' (Chilean) is often used in connection with products, culture, and geography. Typical combinations are 'chilenischer Wein' (Chilean wine), 'chilenische Küste' (Chilean coast), 'chilenische Literatur' (Chilean literature), or 'chilenisches Spanisch' (Chilean Spanish).
Die chilenische Küste ist über 6000 Kilometer lang.
The Chilean coast is over 6,000 kilometers long.
Latin American
'lateinamerikanisch' and 'chilenisch' can both refer to culture, people, or things from Latin America. 'lateinamerikanisch' is the general term for the whole region. 'chilenisch' is more specific and means only Chile. They can be confused because Chile is a country in Latin America. If you do not mean a specific country, 'lateinamerikanisch' is appropriate. If you are talking about Chile, its people, or its culture, 'chilenisch' sounds more natural and precise.

Zum Üben liest der Fahrer nach der Arbeit einen chilenischen Zeitungsartikel.
To practice, the driver reads a Chilean newspaper article after work.
Heute hat ihr Vater ein Buch über lateinamerikanische Geschichte gekauft.
Today her father bought a book about Latin American history.