What he made here is a "Frozen Custard"; It is synonymous with Ice cream so there is little distinction. But iced cream w/o egg is still "Ice cream", but custard is not custard w/o eggs. Ice cream w/o any additional flavoring is an ice cream base (as shown in this ch). Vanilla ice cream is NOT a base for other ice cream. Despite how synonymous vanilla has become with the term plain/boring, it IS it's own flavor.
Fun fact: Ice cream will always form ice crystals as it freezes. The smoothest ice creams are chilled fast while mixing to ensure the crystals form as small as possible.
Sealing the cream, surrounding it w/ salted ice and rolling it around/shaking in a container is the most common low tech technique. I feel like the magic version the twins did would result in clumps of small and large crystals and be less smooth not more.