Modifications of stem

Potato is an underground stem tuber that stores food.
Carrot and sweet potato are modified roots, and groundnut is a fruit.

The potato tuber is a modified underground stem that stores food.
Being a stem, it has nodes and internodes; each 'eye' is a node, seen as a small depression bearing a scale leaf scar.
In the axil of each scale leaf there are buds, so the buds in the eyes are axillary buds.
These axillary buds sprout into new shoots, which is why potato is propagated from tuber pieces carrying an eye.
They are not root buds or flower buds, and 'shoot buds' is only a general term; morphologically they are axillary buds.
Hence the eyes of the potato tuber are axillary buds.

Some plants of arid regions modify their stems into flattened or fleshy cylindrical green structures that contain chlorophyll and carry out photosynthesis; the leaves are reduced to spines to cut down transpiration.
In $Opuntia$ the stem is flattened, green and succulent (a phylloclade); in $Euphorbia$ it is fleshy and cylindrical.
$Casuarina$ has thin, jointed, needle-like green branches, not flat succulent ones.
$Hydrilla$ is a submerged hydrophyte, and $Acacia$ is a tree with normal woody stems.
Hence the plant is $Opuntia$.

An offset is a lateral branch with short internodes, each node bearing a rosette of leaves and a tuft of roots.
It is found in aquatic plants like $Pistia$ and $Eichhornia$; the offsets break off and grow into new plants.
Runners are found in grass and strawberry, stolons in mint and jasmine, and suckers in banana, pineapple and $Chrysanthemum$.
Hence vegetative propagation in $Pistia$ occurs by offset.