There are many things to be discussed on the topic of the "internet", and an exhaustive study of all the connotations and denotations of the "internet" is certainly beyond me, there is one aspect of the internet that I can address: its origins, and the purpose for its introduction.
Ironically, the internet has been a great source of both misinformation and the debunking of that information. And one of these pieces of information is the "urban myth" that the internet was created to survive a nuclear war, as well as the urban myth that that belief is merely an urban myth? Confused yet? Good.
The development of the internet can only be understood in context, and that context is the great development of computer technology between World War II and about 1973, the date of the Mansfield Amendment. Those were confusing years in the United States: there was a combination of peace and prosperity, together with paranoia and preparation for war. The United States government, through agencies including Defense, Energy and NASA, poured money into research, in corporations and universities. This research helped move technology forward, so that computing and information technology became commercially available, and the cycle fed on itself. But the keystone to the development was arguably always national security, and national security in the 1950s was arguably mostly in terms of nuclear war. So the background of any technology during this era was in some ways about nuclear war.
But then we get down to the specifics. The Internet, and even the Arpanet, were not designed all at once, and the specific technologies that make them up were designed at various points for various reasons. Sometimes, those reasons are not very clear.
- Packet Switching, one of the basic technologies behind the internet, and a technology with redundancy as its goal, was first introduced in 1962 for the Rand Corporation. I find the abstract of the paper interesting:
A discussion of the problem of building digital communication networks using links with less than perfect reliability.
I think "less than perfect reliability" is one of the best euphemisms ever for "just had several thermonuclear weapons go off near them". But the original paper on packet switching was more about the idea for reliability, not about implementing it in any immediate circumstances.
- DARPA and what it did: DARPA, or ARPA as it was sometimes called, was the "advanced research projects angency". The military had many different ways of doing research, both internally and externally, and some of these were designed to be specific to a certain service or problem. If the Air Force had been specifically looking for a way to build up redundancy between strategic missile bases, that problem would have been handled by an in-house team, and dealt with a specific problem. The fact that the internet was designed by ARPA means that it was probably more under the category of general research, rather than solving any specific problems.
- Within DARPA, the internet and related technologies were the brainchild of J.C.R. Licklidder, whose stated position was the head of Behavioral Sciences & Command and Control research at DARPA. Licklidder didn't seem to be the type of person to let his job title hold him down, but it seemed he was more interested in psychology and human interactions than technical problems. It seems that his interest in the internet was more in how humans share information, rather than in the technical problems of data transfer.
As I said, there were many other technologies that had to exist before the internet did, and many or most of them came directly or indirectly out of military research. However, the evidence seems to be that the internet was not "built to order" to survive a nuclear war. However, it is inconceivable that military researchers, working in the 1950s and 1960s, could have been working on an intensive communications system and not be thinking about nuclear war. So, as with so much else on the internet, the truth value of the internet's origins and original purpose will remain debatable.