Hello, from my experiences as someone who just use them (headphones, earphones, IEMs) for consumption of music exclusively and acknowledging my complete ignorance in the production side of things, heres what I think of the classification of all things personal audio.

Broadly the classification is based on size, hence headphones and earphones.

Headphones come with a band that sit atop your head and have their drivers surrounded by cushioned pads which sit on or around your ears. On ear headphones usually will have earpads that are just big enough to sit on the edges of the ears and generally do not provide a complete seal around the ears. Around ear headphones have earpads that surround the ears and provide a much better seal, and hence a fuller sound, at least in my experience.The sound is produced by the drivers and the pads ensure the sound is channeled into your ears, and since the sound waves are received on the whole ear, the perception of space and width can be simulated by good design. Another classification that come here will be open back and closed back headphones. The drivers produce sound by creating vibrations in the air by oscillating a thin membrane. The air that faces our ears being vibrated is what we perceive as sound. But when the air inside the driver is functioning like this, the air on the other side goes through the same vibrations. So when a headphone manufacturer is designing a headphone, they have two choices, either find a way to close the space and dampen the unwanted vibrations as much as practicable, or design them with the backs open, with some dampening or more often than not, with just a dust filter to prevent ingress of dust from outside. Both these types of headphones have their own strengths and weaknesses, but generally open back headphones can simulate better perception of space i.e. width and height or soundstage as it is generally known, and closed back headphones can really reproduce bass in a more impactful and detailed manner, among other differences.

Earphones are tinier in size, and their whole construction is designed to sit inside the ear structure, with smaller drivers providing sound into the ear canals directly. The classification here are IEMs and, well, as I’m unsure of what the official terminology is, earbuds or simply non IEMs. The non IEMs are generally just small drivers just sitting in front of the ear canals blasting the sound waves out, not really providing any sort of seal or even good fit. Whereas in the case of IEMs, the drivers sit just outside the ear canals as in the above case, but there will exist a sort of tubing with rubber/foam/silicone tips that would actually get inserted into the ear canal. They provide excellent seal and allow you to listen to your music with as close to total isolation as is practicable, in my opinion.

In the question you posted you mention the cable connector for the IEMs you looked at arent what you would plug into a jack. This is sort of puzzling to me as from what I know, the most common terminations from earphones and headphones are standard, 3.5mm or 6.4 mm. There are other sizes, namely 2.5mm and 4.4mm, but as far as I know these are used for balanced output, which I’m not competent enough to explain. It is true that various manufacturers use different type of cables and terminations from the drivers, some detachable and some fixed, but the termination that is plugged into your source are generally standardised, to the sizes mentioned above, as far I am aware.

This post is super long by now, and there isnt much more I can tell you, apart from just letting you know there exists several types of driver technology, the majority being dynamic drivers, and planar, electrostatic, balanced armature being some of the other ones. Its not within my capability to explain all these, so I’ll leave it to you to look them up if you’re interested.