"Al Gore invented the internet."
"During my service in the United States Congress, I took the initiative in creating the Internet."
"...Gore took credit for the Internet..."
Did Al Gore invent the internet?
The facts of this situation are pretty well known. During his time in Congress, Gore played a vital role in the creation of the computer networking system that we know as 'the internet..
No one person or even small group of persons exclusively "invented" the Internet. It is the result of many years of ongoing collaboration among people in government and the university community. But as the two people who designed the basic architecture and the core protocols that make the Internet work, we would like to acknowledge VP Gore's contributions as a Congressman, Senator and as Vice President. No other elected official, to our knowledge, has made a greater contribution over a longer period of time.
as far back as the 1970s, Congressman Gore promoted the idea of high-speed telecommunications as an engine for both economic growth and the improvement of our educational system. He was the first elected official to grasp the potential of computer communications to have a broader impact than just improving the conduct of science and scholarship [...] the Internet, as we know it today, was not deployed until 1983. When the Internet was still in the early stages of its deployment, Congressman Gore provided intellectual leadership by helping create the vision of the potential benefits of high speed computing and communication.
It's notable that the quote you cite from Al Gore himself is not that he invented the internet, but that he "took the initiative in creating it". Creating something like the internet is not the same thing as inventing it, just as saying that Alexander Graham Bell invented the telephone is not the same as saying he created the telephone network. (It happens that he played a part in the latter also, but the two don't have to be related.)
To summarize: the claims that Al Gore made for himself are largely justified. The truth of claims that other people say he made are dependent on your definition of 'invention', and so subject to opinion.