This is a simplification, but in essence a lot of the confusion over materials arises because 'iron' and 'steel' are not simple things.
Iron is an element. It has quite a high melting point, doesn't corrode quickly but is relatively soft and has very little high temperature strength. Other elements are therefore added to iron to change its properties.
The smelting process (making pig iron from ore) results in a composition with a lot of carbon and silica (melted sand) in it. The carbon alloys with the iron, lowering the melting point and making it brittle (but actually improving the high temperature stiffness). To make it useful, it needs to have some of the impurities removed.
Pre-industrial revolution, the only options were to heat the iron and hammer out the impurities, eventually making wrought iron, or to 'burn' some of them out, making cast iron which still contained a fairly large amount of carbon and hence was still rather brittle, although with excellent high temperature stiffness. Throughout the Victorian era, these were pretty much the only options for large volume material. The development of processes for volume steelmaking e.g. the Bessemer process) changed this in the 20th century.
Steel is an alloy of iron with controlled quantities of other elements, often including carbon at up to ~2%. These elements, such as manganese, nickel or chromium, are selected to improve hardness (e.g. cutting tools), high temperature stiffness, corrosion resistance to heat, seawater, acids etc. Because there are so many possible applications, there are also loads of different steels, optimised to meet them. There are steels designed for continuous service under high pressure at over 750degC without corroding or deforming, which would have your whole stove glowing cherry red! The steelmaking process has become so universal (and so much cheaper) that it has effectively completely replaced wrought iron, through the supply of 'mild steel' which is basically iron with a very small amount of residual carbon.
So, if a stove is made of 'cast iron' you know it will be stiff and hold up well at high temperatures, even if the material is relatively cheap. However if it's too cheap then it may not have been refined enough and hence crack easily. If it's made of 'steel' there is a huge variety available, which at the high end could mean it's much better than cast iron and at the low end means it's much worse. This is where you have to trust the manufacturer.
Not sure if this helps or not!
Alec