Russell was not a proponent of logical positivism, and made his philosophical contributions before the Vienna Circle, Ayer etc.

Russell is associated with the “Direct Reference Theory” due to his influential paper “On Denoting”, which makes the meaning of a name the reference of its descriptive content. In this sense Russell is a “direct reference” theorist, but only in contrast to Frege, who proposed an intermediary “sense” of a name, between the idea and the object to which it refers. Russell’s work did not, at the time, fall under such a name “Direct Reference Theory” although his work may reasonably be associated with this later theoretical development. Furthermore, Russell’s account of number and of mathematics in the Principia Mathematica is not rooted in Direct Reference Theory in the way you describe: they are separate contribution that are only linked by Russell’s broader work on and advocacy of Frege’s logic. Principia Mathematica, as a project, in fact predates Russell’s work in On Denoting.