In his lecture notes on harmonic analysis, Terrence Tao often uses the notion of an adapted bump. A proper definition is given in Definition 8.14 in these lecture notes. However, I don't really get the essence of this definition, in particular because it is not totally clear to me what $O_{k,\Omega}(1)$ should tell me. Also, the kind of "varying" is only explained afterwards and doesn't really satisfy me.
Intuitively, I would have expected that bumps adapted to a ball just are those in which the gradient bounds scale reciprocal with the radius or something like that... Don't sure of this is meant here and how I would read this from the definition.
For an example usage see the last paragraph of page 2 in these notes.