gläubig
devout
A person who has a firm religious faith, especially in a god or the teachings of a religion.
Sie ist eine tief gläubige Christin und geht jeden Sonntag in die Kirche.
She is a deeply devout Christian and goes to church every Sunday.

devout / credulous
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devout
A person who has a firm religious faith, especially in a god or the teachings of a religion.
Sie ist eine tief gläubige Christin und geht jeden Sonntag in die Kirche.
She is a deeply devout Christian and goes to church every Sunday.

The adjective 'gläubig' (devout/credulous) is derived directly from the noun 'der Glaube' (faith/belief). It describes a person who has faith or belief.
Er ist nicht religiös, aber er hat einen festen Glauben an das Gute im Menschen.
He is not religious, but he has a firm belief in the good in people.
You can tell from context whether the religious or the naive meaning is intended. If it is about religion, God, or church, it is the first meaning. In all other cases, often in connection with children or stories, the second, uncritical meaning is intended.
Mein Großvater war ein sehr gläubiger Mann.
My grandfather was a very devout man.
To describe the intensity of religious faith, 'gläubig' (devout) is often combined with adverbs like 'tief' (deeply), 'fest' (firmly), or 'streng' (strictly).
Sie kommt aus einer streng gläubigen Familie.
She comes from a strictly devout family.
In the meaning 'naively trusting', 'gläubig' (credulous) is often used to describe a facial expression. Typical combinations are 'gläubige Augen' (credulous eyes) or 'ein gläubiger Blick' (a credulous look).
Mit gläubigen Augen blickte der Junge zum Zauberer auf der Bühne.
With credulous eyes, the boy looked up at the magician on the stage.
religious
'religiös' and 'gläubig' both describe a connection to religion and faith. 'gläubig' says especially clearly that a person truly believes in God or in the teachings of a religion. 'religiös' is somewhat broader: it can describe a person, but also customs, festivals, or topics. The words can be confused when talking about people. As a simple guide: a 'gläubig' person has faith; something 'religiös' belongs to the area of religion.

Meine Großmutter ist gläubig und betet jeden Abend.
My grandmother is devout and prays every evening.
Sie interessiert sich für religiöse Musik aus verschiedenen Kulturen.
She is interested in religious music from different cultures.