On a web site I have contact details section. This section can contain phone number, email address, personal site, etc. And I have button to add new item for contact details.
What is more correct form for 'Add ...'?
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I think contact information is more common than contact details, although the latter is defined in Collins Dictionary as:
the information required to contact someone, such as an address or telephone number
To see that contact information is more common than contact details, see this ngram.
These two words comprise a collocation and are immediately understood to mean exactly what you want them to mean. Don't be led on the wild goose chase of looking for a single word when two will do just fine.
Add… Contact Method
But go with “Add Contact Details”.
It would be nice if you could just use the single word “Contact”. However, it wouldn’t be proper. That’s why other answers are suggesting “Contact Details” or “Contact Information”.
In the sense of human communication, contact has a few usages.
To say “Add Contact” would imply adding a new entry/person to the directory.
To say “Add Contact Method” implies opening up a dialogue box where you first choose the contact method and then enter the contact details/information.