The question has been asked: “what is a pen?”. The answer is really quite simple. If you use it to (hand)write or draw and it uses ink, it’s a pen. Otherwise, it is (for instance) an ink cartridge or a pencil. As an example of another expression of my same view, Wikipedia’s article on the topic begins as follows:
A pen (Latin pinna, feather) is a long, thin, rounded device used to apply ink to a surface for the purpose of writing or drawing, usually on paper. There are several different types, including ballpoint, rollerball, fountain, and felt-tip. Historically, reed pens, quill pens, and dip pens were used. Modern-day pens come in a variety of colors, shapes and assortments. The most common contain black or blue ink.
It goes on to define a number of types of pens including markers, also known as “felt-tip pens” (of which texters, as I call them here, are a subset). Permanent markers are pens. Whiteboard markers are pens. Even highlighters are pens of a sort!
Note that other things like styli are sometimes considered pens by analogy. This is a metaphoric use of the word and this blog is not required to discuss them.