No, a new edition is when changes are so complete the ISBN registration has to be changed with the government (this is why the new core rulebooks for D&D, if they are anything, are the 9th edition of the PHB & DMG, at least, since they are the 9th set of ISBNs issued for that title).
A print run is when the typesetting has been corrected with errata to correct issues with the prior print runs, but are minor enough changes that the book still has the same ISBN. Ao they can make more books in the first printing at any time, it becomes the second printing when what is being printed has been changed.
Now, I think it is possible they are slow rolling the printing of new books until they get that errata fully sorted, hence the supply issues. Doesn't mean they haven't been printing some, probsvly still trying to figure out how big their market is still.