If I have a database with EU citizens on it, but they provide their American data (American address, phone number, email, etc.), am I bound to GDPR?
This is Art. 3(2) GDPR:
This Regulation applies to the processing of personal data of data subjects who are in the Union by a controller or processor not established in the Union, where the processing activities are related to:
(a) the offering of goods or services, irrespective of whether a payment of the data subject is required, to such data subjects in the Union; or
(b) the monitoring of their behaviour as far as their behaviour takes place within the Union.
So whether someone is an EU citizen, is not relevant. If an EU citizen is in America, the GDPR does not apply.
However, it does apply to American citizens who are visiting the EU, when (a) or (b) above applies. But it only applies for all data processing performed during that visit.
So if someone has an US address and phone number is not enough to determine whether the GDPR does apply. But recital 23 clarifies (a) a bit;
[...] In order to determine whether such a controller or processor is offering goods or services to data subjects who are in the Union, it should be ascertained whether it is apparent that the controller or processor envisages offering services to data subjects in one or more Member States in the Union. Whereas the mere accessibility of the controller’s, processor’s or an intermediary’s website in the Union, of an email address or of other contact details, or the use of a language generally used in the third country where the controller is established, is insufficient to ascertain such intention, factors such as the use of a language or a currency generally used in one or more Member States with the possibility of ordering goods and services in that other language, or the mentioning of customers or users who are in the Union, may make it apparent that the controller envisages offering goods or services to data subjects in the Union.