It's really hard to make generalizations about the naming conventions of various military units, because practices and possibilities differs from country to country, despite there are some trends, or schools (like following German, USian, or Soviet organizational doctrines).
But.
Originally they had the walking infantry with horse towed artillery. Let's say a division had three infantry regiment and an arty note my abstraction breaks here, for example Hungarian military was/is brigade centered not division, the division level is missing, these are/were organized into corps.
Then they invented tanks and wanted tank division and changed a full infantry regiment to a full tank regiment. The rest of the inf and the horse got trucks to roll about so they could follow the tanks, because their speed grew.
Then they saw that "wow those trucked fellas roll pretty quick, what a good way of transport", and motorized units were borned, from full divisions to squadrons. The higher levels of units might got some armour. Armour could mean tank, or some other metal boxes, like recon units with their wheeled tincans, or they added to the division's detached AT unit (eg. battalion or company) a subunit (a company or a squad) of assault guns.
Then they discovered that they could put the infantry into metal boxes too, so they did, they use/used wheeled, tracked, half-tracked vehicles with various thickness of steel platings. This they called mechanized and they switched the motorized infantry of the tank division to this. But they made divisions with only mechanized infantry, however the rest of the subunits were also changed, they could have for example fully armoured AT battalion/company, or they could add a smaller "pure" tank unit into the mix as a detachment for the division (or brigade).
The equipment depended on what they had, could acquire, produce, and even if something is named X it can have Y equipment for a long time still until they get the new stuff. And they are X only on paper.
Also the naming conventions in English for WWII tank units when all these appeared first I got the feeling they use panzer to German, tank to Soviet, and armoured to USian/British ones. In Hungarian we also have this differences, páncélos (=panzer) for German, harckocsi (=armoured car) for Soviet, and tank for Western units. Or something similar.
And I also got the feeling that the word "armoured" in the context of units (and not vehicles) means it contains tanks, or assault guns in some ratio. This is why I would say an armoured unit has heavier weaponry over a mechanized one. Liek (and now this example is pure fantasy) I wanna organize a mechanized division I'd add three regiment of mechanized infatry, a self-propeleld artillery regiment, a self-propelled AT battalion, a self-propelled AA battalion, an armoured recon company, etc etc. But for an armoured division I would add three mechanized infantry regiments but a company in all of the regminets would be a tank company instead, or an detached squadron of tanks or something like that.