Following the discussion in this paper (discussion around Eq. (3) in Page 4) and these lecture notes (discussion in Section 1.2.1 in page 10) given a field theory in some spacetime $(M,g)$ described by an action $I[\phi^i]$ written in terms of a Lagrangian $d$-form in the usual way: $$I[\phi^i]=\int_{M}{\cal L}[\phi^i]$$ a symmetry is defined as any variation $\delta \phi^i$ such that the action changes by a boundary term, or equivalently, the Lagrangian changes by a total derivative: $$\delta{\cal L}=d\Xi\Longleftrightarrow \delta I[\phi^i]=\int_{\partial M}\Xi.$$
My question: what is the motivation to define a symmetry like that allowing a boundary term? Why a symmetry is defined as a variation that might produce a boundary term, thereby giving $\delta I[\phi^i]=\int_{\partial M}\Xi$ instead of a variation that keeps the action invariant giving $\delta I[\phi^i]=0$?
