
die Zusage
acceptance / confirmation
A positive answer, confirmation, or binding consent to a request, offer, or invitation.
Wir haben endlich die Zusage für die neue Wohnung erhalten.
We finally received the acceptance for the new apartment.
acceptance / confirmation
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acceptance / confirmation
A positive answer, confirmation, or binding consent to a request, offer, or invitation.
Wir haben endlich die Zusage für die neue Wohnung erhalten.
We finally received the acceptance for the new apartment.

promise
A firm promise to do or guarantee something specific.
Er gab mir die Zusage, das Projekt bis Ende der Woche abzuschließen.
He gave me his promise to finish the project by the end of the week.
'Die Zusage' (the acceptance/confirmation) is the noun form of the verb 'zusagen' (to agree/confirm). It describes the result of this action: binding agreement or a promise.
Ich kann dir noch nicht fest zusagen, aber ich versuche zu kommen.
I can't firmly agree yet, but I'll try to come.
Common verbs used in connection with a confirmation are 'eine Zusage bekommen' (to get an acceptance/confirmation), 'eine Zusage erhalten' (to receive an acceptance/confirmation), or 'eine Zusage erteilen' (to give approval/confirmation). The preposition 'für' (for) indicates what the confirmation applies to.
Sie hat die Zusage für den Studienplatz in Berlin bekommen.
She got the acceptance for the university place in Berlin.
When the 'Zusage' (promise) is a promise to do something, it is often followed by an infinitive construction with 'zu' (to). This describes exactly what was promised.
Ich habe seine Zusage, mir bei dem Umzug zu helfen.
I have his promise to help me with the move.
word
Both words can express a binding answer or assurance. A Zusage is usually the specific positive response to a request or invitation, while das Wort in fixed expressions places greater emphasis on personal reliability. Use Zusage for confirmation and dein Wort for personal assurance.

Nach der Einladung schickte ich noch am selben Tag meine Zusage.
After receiving the invitation, I sent my confirmation that same day.
Ich gebe dir mein Wort, dass ich an der Reise teilnehme.
I give you my word that I will take part in the trip.