notieren
to note
To briefly write something down so as not to forget it or in order to document it.
Bitte notieren Sie sich die Telefonnummer für den Rückruf.
Please note down the phone number for the callback.

to note / to be quoted
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to note
To briefly write something down so as not to forget it or in order to document it.
Bitte notieren Sie sich die Telefonnummer für den Rückruf.
Please note down the phone number for the callback.

The verb 'notieren' (to note) belongs to the same word family as 'die Notiz' (note). Both words have the same root and refer to writing down information.
Ich mache mir immer Notizen, damit ich nichts Wichtiges vergesse.
I always take notes so that I don't forget anything important.
The meanings are easy to distinguish: when a person 'notiert' something, they write it down. When a stock or security 'notiert', its current stock market price is being stated.
Die Sekretärin notierte die wichtigsten Punkte der Besprechung.
The secretary noted the most important points of the meeting.
'Notieren' (to note) is very often used with a dative pronoun (mir, dir, sich). The construction 'sich (dative) etwas notieren' means that you write something down for yourself.
Kannst du dir bitte meine neue Adresse notieren?
Could you please note down my new address?
This meaning of 'notieren' (to be quoted) is technical language from finance and stock exchange contexts. In everyday life, the word is used almost exclusively to mean 'aufschreiben' (to write down).
Der DAX notierte am Mittag leicht im Minus.
The DAX was quoted slightly lower at midday.
to stand
'stehen' and 'notieren' can both have to do with written information. 'stehen' describes that something is already written in a text, on a sign, or in a message. 'notieren' describes the action: someone briefly writes something down so they do not forget it. Learners may confuse the words because in both cases something written is involved. The simple difference is: information 'steht' somewhere; a person 'notiert' information.

Bitte notiere dir die Telefonnummer gleich im Handy.
Please note down the phone number in your phone right away.
Auf dem Zettel steht die neue Adresse der Ärztin.
The doctor's new address is written on the note.