It is difficult, if not impossible, to determine which "recent American book" Arnold is referring to, but the statement he makes is certainly correct.
There are some very interesting surveys about the history of the concept of manifold, for example in The Concept of Manifold, 1850-1950
we read (p. 15)
During the 1880s Poincare came across "manifolds" in several analytical or geometrical contexts, although he personally understood them at that time still in a rather vague way.
and (p. 16. emphasis mine)
Poincare gave a discussion which in fact spoke in favour of the surjectivity and was already sharper than Klein's, but still used highly intuitive ideas about continuous variation of images in higher dimensional spaces in a symbohcally uncontrollable manner. Even the spaces themselves were not shown to be manifolds but taken as such, without further ado. For any critical reader (perhaps even including Poincare himself) the "continuity proof" could thus be taken at least as much as an indicator for the necessity of an
improved understanding of higher dimensional geometry as it was an indicator for the truth of the uniformization theorem. And in fact a clarification of the topological proof strategy was given only later by Brouwer [...]
Veblen and his student J.H.C. Whitehead, coming from (and going back to) Oxford, brought the axiomatization of the manifold concept to a stage which stood up to the standards of modern mathematics in the sense of the 20th century
The axiomatization by Veblen and Whitehead was first exposed in A Set of Axioms for Differential Geometry, and then in The Foundations of Differential Geometry.
Coming back to the book Arnold refers to, one possibility could be A History of Algebraic and Differential Topology, 1900-1960 by Jean Dieudonné, published in 1989 (Arnold's paper is from 1998), but I highly doubt that Arnold would consider it an " American book".