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Can't confirm from dictionaries, but Nils' case is good and WWW pages and images support it. Reading is correct, it seems.

I’m not completely clear on the distinction between 子芋 and 孫芋 – simple definition I’ve given is probably fine for all except specialists though.

The 親芋、子芋、孫芋 hierarchy is clear enough naming-wise, so I’ve rendered the latter as “secondary” and “tertiary”, which conveys the feel, but I’m not sure these are clearly distinguished in English (the terms “cormel” and “cormlet” seem to be used interchangeably, though I’m admittedly not an agronomist; I’ve roughly interpreted “cormlet” as being the smaller one, if push comes to shove).

ja WP doesn’t seem to clearly distinguish them in the text, and I can’t really make out the picture; there may be a distinction that I’m missing.
Some distinction seems to be made between small *basal* corm growths and ones from the side, but I’m not sure whether this corresponds to the naming.
It might also just be a matter of being this year’s new corm(let) versus one that’s a year old already.