Current definition: A branch that forms from a leaf at the very base of the shoot.
Basal axillary shoot is a subclass of axillary shoot system ("A shoot-borne shoot system that develops from an axillary vegetative bud."), not a branch. Also, it does not develop from a leaf, but in a leaf axil.
New proposed def.: An axillary shoot shoot system (PO:0006343) that develops from an axillary vegetative bud (PO:0004712) that is part of a stem base (PO:0008039).
Comment: Called a tiller in the Poaceae.