Re #1: You can of course say 'database instance', but then you risk that someone might think you mean 'DBMS instance' which is different as it also includes the software needed to manage the database. Besides, my definition is supported by SQL - is that formal enough for you?Ummmm... in your post, #1 can be described as a database instance (https://en.wikipedia.org/wiki/Instance_(computer_science)) on a server. Actual tables, and related programs to make use of contents. #2 can be described as a dataset. (https://en.wikipedia.org/wiki/Data_set) These are actual terms with a formal definition.
SQL:
CREATE DATABASE foo;
Re #2: Agreed, 'data set' is probably a better term for this. Sadly, that doesn't prevent people from using/misusing "database" to describe the same thing, though!