Free morphemes do not require other morphemes to make sense. That means that all free morphemes are words. Content words have meaning, but no function beyond that meaning: examples include dog, house and car.
The category of functional words is comprised of all conjunctions, prepositions, determiners, auxiliary verbs, modals, qualifiers, question words and pronouns, who serve a function instead of possessing a concrete meaning. Examples include the, over and her. Content words is an open class of words, meaning it receives additions more commonly. Functional words is a closed class of words, meaning it rarely receives additions.
Bound morphemes require other morphemes to make sense. Therefore, a bound morpheme is either a root or an affix. Roots can be both bound morphemes and free morphemes. Roots are just the remnants after all affixes have been removed. If the remnant root doesn't make sense on its own, then it is a bound root. If it does make sense, it is a word, and a free morpheme. Examples of bound roots are -ceive and sci-.
Affixes are additions to a word, either at the front (prefix), end (suffix), in the middle (infix), around (circumfix), at multiple places (transfix). These additions may take the form of one or multiple phoneme changes (simulfix), the full or partial, identical or similar, duplication of a root/stem/word (duplifix) or the removal of a part of the word (disfix).
The derivational affixes modify the word's meaning. Examples include pre-, post-, dys- and mal-.
The inflectional affixes modify the grammatical properties of the word, such as a verb's tense, aspect, person, mood or number, or a noun/adjective/pronoun's number, gender or case. According to Wikipedia, affixes that change the class of a word (comprised of nominal, verbal, adjectival and adverbial affixes) are a part of the derivational affixes. This doesn't quite make sense to me, as I thought a word's class was a grammatical property.
Empty morphemes are phonemes that add no meaning to the word. If an empty morpheme is also an infix, the morpheme is called an interfix. All examples of empty morphemes in the English language that I know of are interfixes: -o- in speedometer, -u- in factual and -u- in sensual. Given how a morpheme is the smallest meaningful unit of language, it seems like empty morphemes are by definition not morphemes. Yet, Wikipedia's article on Bound and free morphemes mentions empty morphemes. In Norwegian, there is sometimes an empty morpheme in between two words, like in arbeidsliv (arbeid [work] + s + liv [life]). This happens in German too. However, in all these cases, the interfixes would be better described as a phoneme, and not a type of morpheme. How come anyone bothered coining the term "empty morpheme"?
Null morphemes are morphemes without phonemic content. In English, there is no affix for words that are singular. No visible or audible affix, that is. There has been conceptualized an affix, or a null morpheme. For example, dogs can be divided up into dog + s, the s being the inflectional morpheme that changes the root's number. However, dog could be divided into dog + /Ø/, the /Ø/ being an unpronounced and unwritten morpheme that may only exist in the average person's subconscious conceptualization of words and the linguistically educated person's subconscious and conscious conceptualization of words.
Now, another way of dividing up morphemes is into lexical morphemes and grammatical morphemes, also called content morphemes and functional morphemes, which is confusing due to the categories of content words and functional words. This confusion is the biggest component of uncertainty in my understanding of morpheme typology. This is how this other categorization looks.
The category of empty morphemes doesn't really have a place on this diagram because they neither have meaning nor a grammatical purpose. If they should exist on the first diagram is debatable in the first place.
So, the questions I am left with are these: Are the two featured categorizations correct? Why aren't class-changing affixes regarded as inflectional affixes? What is really the fundamental difference between a bound root and an affix? They are both units of meaning that require other morphemes to make sense; why is one considered a root and the other not? Why are "empty morphemes" a thing when they don't comply with the definition of morphemes?