Are beets considered tubers?

No, beets are taproots, sweet potatoes are lateral roots (tubers). Potatoes are modified stems (stem tubers).

In a plant with a taproot system, the taproot is the largest, most central, and most dominant root. Typically a taproot is somewhat straight and very thick, is tapering in shape, and grows directly downward. In plants with taproot systems, the taproot is the central root from which other roots sprout laterally.

Beets, carrots, radishes, and turnips are taproots.
https://en.wikipedia.org/wiki/Taproot

A tuberous root or storage root, is a modified lateral root, enlarged to function as a storage organ. The enlarged area of the root-tuber, or storage root, can be produced at the end or middle of a root or involve the entire root. It is thus different in origin but similar in function and appearance to a stem tuber. (Potatoes are stem tubers.) Examples of plants with notable tuberous roots include the sweet potato.

https://en.wikipedia.org/wiki/Tuber

Root crops and tubers have a different kind of root structure. Root crops are tap roots. They are a single root that bores down into the soil like a carrot or parsnip. They can be a single rounded modified root like a beet. The key point being, root crops form one vegetable per stem. With tubers, a single plant can produce several tubers. A carrot plant will have one carrot, while a potato plant can produce several potatoes.

https://lifeandagri.com/root-and-tuber-crops/#:~:text=All%20tubers%20are%20root%20crops.%20A%20single%20plant,Tubers%20offer%20loads%20of%20energy%20%28starch%29%20and%20vitamins. (new link, satguru)