I noticed that some really stinking old books that are on the public domain sometimes have a newer copyright notice from the publishing company. Let's say the book was written on 1820 and it has a 1980 copyright notice from such and such publishing company. Does that mean that the book is no longer on the public domain?

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A book is put into the public domain 70 years after the author has died. This basically means the text is out of copyright and free to read or reproduce. However, there are certain notable exceptions to this. Often an ammended version of the book is released and by doing this the book is put back into copyright and so out of the public domain. So, let's say author X releases a book. 50 years after his death it is discovered that a certain word in the original publication was misprinted then the publisher would reissue the text and copyright would begin again. Some shrewd publishers are able to do this again and again. This has happened with the Hemmingway estate and Emily Dickinson. Both are still in copyright and so not in the public domain for many years to come eventhough their initial 70 year copyright protection is over. The upshot is we can't get them for free online and have to pay for the book copy.

That does indeed mean the book is no longer in public domain. As long as someone holds the copyright, it isn't PD