Those were the historical and technical divisions.
Because http & https is ubiquitous and they have a defacto pass through most firewalls, the modern "web" is the "internet" as far as application interfaces are concerned.
The idea of a pure separation of concepts such that http is "documents" and some other protocol on some other port is "apps" is not going to happen. All those IT departments' security teams and Cisco admins at Fortune 1000 companies are not going to adjust their firewall rules to open new ports and protocols. The "http port 80" has become the "internet" on top of the old definition of "internet". It's not ideal but that's where we are today.
I think we need another rule of thumb here: All standards will be modified by public cluelessness, and exist as uglier de facto distortions of themselves.
I don't see how willfully blending these two things is helpful for anyone.
In this context, it can be safe to assume that everyone knows this technical distinction (even for those of us who are not web programmers). In other contexts with less computer-technical people, I don't think being annoying and correcting every "technically incorrect" mention of "The Internet" when they mean The Web is helpful. But I think it is helpful to remind them of the distinction if you are actually talking about the subject (as opposed to just saying things like "look at your internet, I just sent you a facebook message"). Computers are magical enough as it is; it doesn't help anyone to make it even more so when they are even slightly interested in discussing some part of computing or IT.
Explaining that The Internet is the infrastructure and the Web is just one the things that uses this infrastructure seems simple enough to grasp, and isn't an oversimplification at all. People understand that the electricity grid has to exist in order for them to make toast.
Because http & https is ubiquitous and they have a defacto pass through most firewalls, the modern "web" is the "internet" as far as application interfaces are concerned.
The idea of a pure separation of concepts such that http is "documents" and some other protocol on some other port is "apps" is not going to happen. All those IT departments' security teams and Cisco admins at Fortune 1000 companies are not going to adjust their firewall rules to open new ports and protocols. The "http port 80" has become the "internet" on top of the old definition of "internet". It's not ideal but that's where we are today.