The best way to avoid confusion is to extend the term Page Rank when you are using it in a conversation or article to indicate what you are referring to when you say "Page Rank". Use the terms "Google PageRank" and "Page Ranking" or "Ranking of a Page" instead.

But what exactly are the two meanings of the term and why is one of the two supposed to be inaccurate or incorrect most of the time, when displayed via a certain very popular browser plug-in or specialized SEO tools?

"Page Rank" (two words), incorporates ALL Ranking criteria used by a Search Engine Algorithm to calculate the Value a page has for a specific Search Term. All Pages that match a Key Phrase are sorted by the Page Rank Value.

This is part of every Search Engine that displays search results by "relevance" and not just alphabetically.

"PageRank(tm)" (PR) - the Google PageRank, is a numeric value that represents how important a page is on the Web. PageRank is a trademark Term of Google and the algorithm named PageRank is patented by Sergey Brin and Lawrence Page. Google figures that when one page links to another page, it is effectively casting a vote for the other page. The famous PageRank algorithm allows the determination of each pages PageRank in relationship to all pages available on the Web (in the Google Index).

The Google PageRank is only one of over 100 criteria for Google to determine the "Page Rank" of a Page for a specific Key Phrase in Google's vast Search Index.

Google Toolbar PR vs Actual PR The Google PageRank(tm) value displayed in the Google Toolbar (the value between 0-10) is not the value used by Google Internally. That means that for example a Page where the Toolbar shows "0" as PageRank might has internally a much higher PageRank, between 3 and 4 for example. The reason for the difference is not clear and people can only speculate.

Even if the displayed PageRank Value would be somewhat accurate, so is it still not the exact number/value Google is using in their Algorithm.

The actual calculated number using the original algorithm is a floating point number with a lot of digits after the decimal point. A site with PR 1 could have an actual PR of 1.298245 another one with PR1 might only has an actual PR of 1.00123.

Integer vs Floating Point The fact that the Toolbar value is rounded to the next whole number is important, because this invisible difference is making a bigger difference the higher the PageRank of a site becomes (a difference of hundreds of thousands of inbound links between 2 pages with the same displayed PR).

Furthermore, the "Google PageRank(tm)" is a number/value assigned by Google to a page/document, independent from the Keyword Phrase. "Page Rank" on the other hand is the number/value assigned to a page in combination with a specific Keyword Phrase.

Conclusions

Fact 1. "Page Rank" is not the same as "Google Pagerank?."

Fact 2. "Google Pagerank? shown in the Google Toolbar is not the same as the actual "Google Pagerank?" number or value used by Google to determine the "Page Rank" of a page for a specific key phrase.

Fact 3. "Google Pagerank?" value is for a specific page or document on the web and the "Page Rank" value is for a specific web page/document and keyword phrase combination.