Cell decompositions have been used in topology for a long time as a tool in computing cohomology, but the notion in algebraic geometry and arithmetic geometry of *paving by affine spaces* (or "affine paving" for short) seems to be a little more recent. An early occurrence is found in the survey Barry Mazur gave in 1974: *Eigenvalues of Frobenius acting on algebraic varieties over finite fields*, Algebraic geometry (Proc. Sympos. Pure Math., Vol. 29, Humboldt State Univ., Arcata, Calif., 1974), pp. 231–261, Amer. Math. Soc., Providence, R.I., 1975.
He considers only complete varieties (where the most natural examples arise),
but the definition makes sense for any variety $X$: require $X$ to have a
filtration $0=X_0 \subset X_1 \subset \dots \subset X_d = X$ with each $X_i$ closed and with $X_i - X_{i-1}$ a finite disjoint union of copies of affine space $\mathbb{A}^i$. (Though not stated explicitly, the union here might be empty.)

This definition was used by Hotta and Springer in their influential 1977 paper here. But as time went on, a broader definition seems to have taken over: given a filtration of $X$ by closed subvarieties as above, one only requires that each $X_i - X_{i-1}$ be a finite disjoint union of copies of a single affine space (of some dimension).

It's not clear to me whether these two definitions (narrower and broader) are actually interchangeable, but the underlying notion of affine paving is itself quite natural. The prototype is $X=\mathbb{P}^1$, with a closed point and a complementary affine line. More generally, a *flag variety* $G/B$ has a suitable filtration with differences given by unions of Bruhat cells $BwB/B$, each isomorphic to affine space of dimension $\ell(w)$ as $w$ ranges over the Weyl group. One can then try to form an affine paving of a Springer fiber (or more general Hessenberg variety) inside $G/B$ by simply intersecting with a Bruhat-type filtration. Work in this direction occurs in a number of places, including $\S11$ of J.C. Jantzen's monograph *Nilpotent orbits in representation theory*, contained in the 2004 Birkhauser volume *Lie Theory* (Progress in Mathematics 228). There are also research papers by T. Haines (on Kazhdan-Lusztig "purity"), J. Tymoczko, E. Sommers, M. Precup (on Hessenberg varieties), etc.

Given the range of applications in the literature, should the broader or narrower definition of "covering by affine spaces" be regarded now as standard?

To me it seems awkward to leave ambiguity in a widely used mathematical definition. But it also seems unnatural to use a broader-than-needed definition.