It depends on what definition of "type system" you're using. Colloquially many programmers use it to refer to any system that checks whether objects have specific shapes. Academics, on the other hand, have a very specific definition of a type system that excludes dynamic detect languages. From TAPL (one of the authoritative works on the subject):

> A type system is a tractable syntactic method for proving the absence of certain program behaviors by classifying phrases according to the kinds of values they compute.

And later on:

> A type system can be regarded as calculating a kind of static approximation to the runtime behaviors of the terms in a program. ... Terms like "dynamically typed" are arguably misnomers and should probably be replaced by "dynamically checked," but the usage is standard.

In other words, you're both correct in your definitions depending on who you're talking to, but if we're going to get pedantic (which you seem to be) OP is slightly more correct.

Personally, it feels like dynamically typed language advocates have been getting more and more vocal about their language of choice being "typed" as static typing has grown in popularity in recent years. This seems like misdirected energy—static typing advocates know what they're advocating for and know that dynamically typed languages don't fill their need. You're not accomplishing much by trying to force them to use inclusive language.

Rather than trying to push Python as a typed language it seems like it would be more effective to show why dynamic checks have value.