Changes to Series[...] in version 11.1

There is an undocumented function System`SeriesDump`truncateSeries that you can use to facilitate the truncation of your series.

System`SeriesDump`truncateSeries[Series[x^2 + x^3, {x, 0, 0}], {x, 0, 0}]
(* O[x]^2 *)

Modify the Series function so that it calls System`SeriesDump`truncateSeries after it is applied:

ClearAttributes[Series, Protected];
Series[expr_, {var_, x0_, nf_}] := 
  Block[{$inSeries = True, result},
    result = System`SeriesDump`truncateSeries[
      Series[expr, {var, x0, nf}], {var, x0, nf}];
    result] /; ! TrueQ[$inSeries];
SetAttributes[Series, Protected];

Then,

Series[x^2 + x^3, {x, 0, 0}]
(* O[x]^2 *)

All disclaimers associated with using undocumented functionality and modifying built-in functions apply.