There is nothing wrong with how that shows up. For example, My windows 7 machine that i'm typing to you on shows that I am connected from my machine to "multiple networks" (Public Networks) and then out to the Internet. Its also possible that your router will show up to your computer as a public network. Does this change its abilities to provide you with the network access you need? of course not. 
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Re: Question regarding "Network Internet access"
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