Re: If you replace the haft and the blade is it the same axe?
tl;dr - Original Atari Inc. split in mid-80s after videogame crash, its two "successors" are both themselves long defunct, PCs weren't a major factor, and today's "Atari(s)" are just what used to be Infogrames.
Longer reply- Bits of that are right, but lots isn't.
The original Atari Inc. did well in the *early* 80s... right up until the 1983 North American video game crash hit and Warners' former golden goose started haemorrhaging money. *That* was what brought about the end of the "original" (and some would argue only true) Atari.
In 1984 Warner sold off the computer/console division to Jack Tramiel- forming the basis of his "Atari Corporation"- leaving behind the arcade division ("Atari Games"), which was sold completely separately the following year.
This site suggests the PC line came out in 1987, i.e. under Tramiel's ownership. I think it was a red herring, rather than the reason for even Atari Corporation's downfall- I don't recall it being that big a deal at the time, when Atari Corp. was enjoying success in Europe with the Atari ST.
It was more the decline of the ST and the failure of their later products (including the Jaguar console) that brought about Atari Corp's enforced demise in the mid-90s when (as you suggest) it had lots of money from litigation, but no products or future. It effectively died after its "merger" with JTS (a second-rate HDD manufacturer), which was little more than a mechanism for Tramiel to reinvest Atari's monetary value in JTS.
One can argue about how much of a "true" successor Atari Corporation was to the original Atari Inc. They continued the existing products, but got rid of most of the existing staff and the "Power Without the Price" approach of the Atari ST, shoestring operation and general philosophy of Tramiel's Atari was completely different to that of Atari Inc. under Warner's ownership.
Regardless, both Atari Corp. and Atari Games are themselves long defunct, and anything after that *is* just exploitation of the name and IP. As far as I'm aware, the current Atari(s) are just the descendants/subsidiaries of Infogrames, which bought the rights in the early 2000s and renamed itself "Atari".