der Aufschrei
cry
A loud, sudden cry that expresses pain, fear, shock, or surprise.
Als er die Maus sah, stieß er einen kurzen Aufschrei aus.
When he saw the mouse, he let out a short cry.

outcry
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cry
A loud, sudden cry that expresses pain, fear, shock, or surprise.
Als er die Maus sah, stieß er einen kurzen Aufschrei aus.
When he saw the mouse, he let out a short cry.

'Der Aufschrei' (the outcry/cry) is the nominalization of the verb 'aufschreien' (to cry out). The prefix 'auf-' here emphasizes the sudden, loud beginning of the crying out.
Ihr plötzliches Aufschreien erschreckte alle im Raum.
Her sudden crying out startled everyone in the room.
The literal meaning refers to a short, loud cry from a single person. The figurative meaning describes a loud, public protest reaction from a group or society.
Er unterdrückte einen Aufschrei des Schmerzes.
He suppressed a cry of pain.
In the literal meaning, the verb 'ausstoßen' (to let out/utter) is often used: 'einen Aufschrei ausstoßen' (to let out a cry).
Als sie die Spinne sah, stieß sie einen lauten Aufschrei aus.
When she saw the spider, she let out a loud cry.
In the figurative meaning, 'Aufschrei' (outcry) is often combined with verbs like 'sorgen für' (to cause) or 'auslösen' (to trigger). A common phrase is also 'ein Aufschrei der Empörung' (an outcry of indignation).
Die Entscheidung der Regierung sorgte für einen Aufschrei in den Medien.
The government's decision caused an outcry in the media.
protest
'der Protest' and 'der Aufschrei' can both describe a public reaction against something. 'Protest' sounds more planned and visible: people openly show that they reject something, for example in the street or with a signature. 'Aufschrei' emphasizes the strong outrage of many people. It is often sudden and emotional, for example after a news report or decision. The words can be confused because both express rejection. If people actively do something, 'Protest' fits better. If the main idea is strong public outrage, 'Aufschrei' fits.

Nach der Entscheidung gab es einen großen Aufschrei in den sozialen Medien.
After the decision, there was a major outcry on social media.
Der Protest der Studierenden richtete sich gegen die Schließung der Bibliothek.
The students' protest was directed against the closure of the library.