You are talking about one where the goal is to make querying the database more idiomatic on your development language, at the cost of some flexibility. This works very well as long as you stay within the bounds of the abstraction, and breaks terribly when you step out of it. The engineering goal is to make the abstraction just broad enough to represent most of the common queries without making it less idiomatic.
The second kind is the type that tries to abstract databases into a specialized query language. The goal here is to bring things you don't get on plain SQL (like type integration or a single DBMS independent language) without losing expressive power. That's the one the GP is talking about.
Maybe you mean something else, but I haven't had any issues when I've had to break out of the ORM and write portions in SQL. (usually once or twice every couple of development man years)
I'm not sure what type integration is, but the ORM I'm most familiar with does allow spanning multiple DBMS's with the same code. (unless when you had to break into DBMS specific SQL for performance reasons)
Is type integration allowing static type checks against your query language? Entity Framework does this as well.
I think EF is the second type of ORM, unless I'm misunderstanding you.
You are talking about one where the goal is to make querying the database more idiomatic on your development language, at the cost of some flexibility. This works very well as long as you stay within the bounds of the abstraction, and breaks terribly when you step out of it. The engineering goal is to make the abstraction just broad enough to represent most of the common queries without making it less idiomatic.
The second kind is the type that tries to abstract databases into a specialized query language. The goal here is to bring things you don't get on plain SQL (like type integration or a single DBMS independent language) without losing expressive power. That's the one the GP is talking about.