I went through a similar process of trying to understand books teaching these same three configurations.
(Also, at that time, I didn't know about single-ended vs differential modes for signalling. The idea of single-ended signalling was assumed by those writing the texts I read. They wrote that a signal arrives on one wire, gets amplified in some way, and leaves on another wire. They then wrote that since the transistor is a 3-terminal device, there are three possible ways to apply one input and one output. That made logical sense. But I still didn't really understand much, except that they were right about the combinatorics.)
Looking back, none of that helped me to understand circuits. It was technically correct -- like a dictionary is technically correct. But it still left me without any idea why any of it mattered. I could parrot the words. But I didn't really understand. Not really.
Learning about these three configurations is more about syntax(the rules governing arrangements) than it is about semantics(the meaning of those arrangements).
And meaning is what matters.
When learning a new language, one must learn some syntax. But one learns faster and much better if also immersed in meaningful situations, and associated language semantics, allowing that process to deepen syntactic abilities. Spending too much time on syntax, before being exposed to why, unnecessarily slows acquisition of knowledge.
For example, a class-A cascode arrangement uses the common base configuration:
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simulate this circuit– Schematic created using CircuitLab
Among the three possible classifications, \$Q_1\$ is in a common-base arrangement. But that doesn't tell you anything about why. You know the syntax. But you don't know the meaning.
So why is that common-base arrangement of \$Q_1\$ important? Well, the Early Effect of \$Q_2\$ is almost entirely mitigated because \$Q_2\$'s collector voltage only varies by a very small amount as the input signal varies. This means the Miller Effect is drastically reduced and therefore the frequency/bandwidth capability of the result is greater. It also enhances linearity, increase gain, and may be used to increase the output impedance (where that matters.) It costs complexity, limits input and output range, and adds power dissipation, as costs. (See some added thoughts on cascodes here.)
But the point I'm hoping to make clearer is that you should focus on various topologies and their purposes and function, and spend less time dwelling excessively on their syntax. Sure, a cascode can be classified as "common-base." But that's an incidental fact. Knowing that classification doesn't help you understand. It's just a name for a thing.
Dwell on various uses and let the syntactic details work themselves out as you go.
There's a parable of sorts on this subject in "The Beat of a Different Drum," by Jagdish Mehra. (It's a biography on Richard Feynman and in my opinion it's the best one to read if don't mind being exposed to the mathematics.)
The following is fair-use text from Mehra's biography:
In the summer the Feynmans would take their vacations in the Catskillmountains. There would be a large group of people there, but thefathers would all go back to New York to work during the week and onlycome back again over the weekend. 'On weekends, when my father came,'recalled Richard, 'he would take me for walks in the woods. When theother mothers saw this, they thought it was wonderful and that theother fathers should take their sons for walks. They tried to work onthem but they did not get anywhere, at first. They wanted my fatherto take all the kids, but he didn't want to because he had a specialrelationship with me. So it ended up that the other fathers had totake their children for walks the next weekend.
'The next Monday, when the fathers were back at work, we kids wereplaying in a field. One kid said to me, "See that bird? What kind ofbird is that?" I said, "I haven't the slightest idea what kind ofbird it is." He says, "It's a brown-throated thrush. Your fatherdoes not teach you anything!"
'But it was the opposite. He had already taught me: "See that bird?It's a Spencer's warbler." (I knew he didn't know the real name.)"Well, in Italian it's Chutto Lapittida. In Portuguese, it is Bom daPeida. In Chinese, it's Chung-long-tah, and in Japanese it is KatanoTekeda. You can know the name of that bird in all the languages inthe world, but when you are finished, you'll know absolutely nothingwhatever about the world. You'll know about the humans in differentplaces, and what they call the bird. So let's look at the bird andsee what it is doing -- that's what counts." I learned very earlyfrom my father the difference between knowing the name of somethingand knowing something.'
Feynman explained, 'My father understood that knowledge was differentfrom the names of things. The names of things are only a conventionthat human beings use to discuss things, and of course that isimportant. But when he would tell me about looking at the birds, itwas not just to look at them but to see what they were doing. As anexample, he said, "Look, see the birds walking around there. Theyseem to be pecking their feathers all the time. Why do you think theydo that?" And I said, "Well, I don't know." I was a kid of ten oreleven. I said, "Maybe their feathers get ruffled when they areflying." I made an attempt at an explanation. He then said, "If thatwere the case, they would peck more when they just landed after theyflew. And after they got straightened out, walking around, theywouldn't peck so much. So let's see, watch those that land and thensee how long they go on pecking and whether or not they peck in theirfeathers at the same rate." After a while we discovered that indeedthey did. So it was not due to a need to straighten out theirfeathers just after flying. You see, he had made a little experiment,learning how to observe and discuss.'
The point I'm hammering here is that you should focus on observing behavior and avoid getting mired by the names we use.