Well what stuff is called, probably depends on your language.......
It seems that I'm the only english speaker here ATM. Whereas you lot speak American. If you want to split hairs, American english. Either way, it is a different dialect.
In current parlance, the original picture vessel is the correct shape for what's often termed as demi-john, except it would need two finger loops molded on and be either a half or gallon capacity (that'd be real gallons not those short measures you lot put up with :: ).
The ones shown as balloon shaped with plastic baskets, would generally be thought of as carbouys. Historically they would have had wire mesh baskets lined with straw, the plastic baskets are the updated ones.
Yes carbouys would have contained hazardous material like acids etc. Demi-johns would have generally contained potable liquids......
Yet as the quoted stuff mentioned, the names can be and are interchangeable. So it doesn't really matter does it......
Just get the damn thing cleaned, sanitised and full of something to ferment......