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Signo de exclamación vs. signo de admiración vs. punto de exclamación

While signo de exclamación, signo de admiración, and punto de exclamación all refer to the punctuation marks ¡ and !, their usage depends on grammatical correctness, history, and foreign influence. The Royal Spanish Academy (RAE) prefers the first term because not all exclamations denote admiration.

Signo de exclamación

A2
This is the current, standard technical term preferred by the Royal Spanish Academy (RAE). It accurately describes the function of the punctuation marks ¡ and !, which is to represent an exclamation (shouting, emphasis, or strong emotion) rather than just admiration.
En español es obligatorio escribir el signo de exclamación de apertura.
(In Spanish, it is mandatory to write the opening exclamation mark.)
La profesora nos explicó que el signo de exclamación indica un tono de voz fuerte.
(The teacher explained to us that the exclamation mark indicates a loud tone of voice.)
Debes poner un espacio después del signo de exclamación de cierre.
(You must put a space after the closing exclamation mark.)

Signo de admiración

B2
This is the traditional name for the symbol, widely used by older generations and in older textbooks. Historically, these marks were called notes of admiration because they frequently expressed wonder or surprise, though this term is now considered less precise than signo de exclamación.
Mi abuelo todavía llama a estos símbolos signos de admiración.
(My grandfather still calls these symbols signs of admiration.)
En la poesía antigua, el signo de admiración se usaba para mostrar sorpresa.
(In ancient poetry, the sign of admiration was used to show surprise.)
El diccionario define estos caracteres como signos de admiración o exclamación.
(The dictionary defines these characters as signs of admiration or exclamation.)

Punto de exclamación

B1
This term is often viewed as an anglicism (a loan translation from the English exclamation point) or a reference specifically to the closing mark. While understood, purists often avoid it because punto implies a single dot at the end, whereas Spanish grammar treats the punctuation as a two-part signo (sign) encompassing both ¡ and !.
En el teclado, el punto de exclamación está encima del número uno.
(On the keyboard, the exclamation point is above the number one.)
No termines cada frase con un punto de exclamación.
(Do not end every phrase with an exclamation point.)
El diseñador pidió que el punto de exclamación fuera más grande.
(The designer asked for the exclamation point to be larger.)

Summary

Use signo de exclamación for proper, modern grammar. You will hear signo de admiración in traditional contexts or from older speakers. Punto de exclamación is often used casually or when referring to computer keys, influenced by English.