I think you're on the right track.

Vulgar Latin is Latin as it was spoken by the people ("on the streets" so to speak). We can call it equivalently "Proto-Romance", because it's the ancestor to the Romance languages, and because by definition "Proto-" languages are reconstructed and not directly documented, which is for the most part true of Vulgar Latin also (a little quoted usage in some literature, some engravings here and there, but still probably influenced by the written standard). Already some changes that are not apparent in written Latin had occurred at the time of a unified Vulgar Latin / Proto-Romance, because they're reflected in all daughter (Romance) languages.

Latin as we know it is very formal and stylized. No one spoke like Caesar's writing. But still obviously it's closely related to whatever preceded the Romance languages.

So it's a bit misleading to say they don't come from Latin. It's more accurate to say they don't come from written Latin, which is all we have records of.

Written languages tend to be somewhat archaic. So what's going on here is that the Romance family includes the modern daughter languages and Proto-Romance, which is itself a daughter of earlier Latin, the basis for formal written Latin. You can think of Proto-Romance and (written) Latin as closely related sister languages. Still, this is more of a technical point regarding reconstruction, etc. There's no real doubt that what the Romans spoke (broadly "Latin") became the modern Romance languages. After all, it was spread by Roman soldiers, citizens, etc.

Another perspective to separate them might be that Latin has remained timeless to some degree, and was used as an international written standard until a few centuries ago. In that sense, the Romance languages were changing while "Latin" was not. And the most extensive continued use of Latin was in Romance-speaking countries. There are lots of cases of borrowed words in Spanish, for example, from Latin. So that's another way to think of Latin and Romance as being parallel, like sisters, rather than one being a direct descendant of the other.

In terms of the broader grouping:
Proto-Indo-European split into numerous sub-families (Germanic, Slavic, Indo-Iranian, etc.), including Italic.
Italic consisted of Latin and several other languages spoken in or near the Italic peninsula.
Latin had several varieties including written Latin (which itself had several varieties / variation over about 2000 years) and Vulgar Latin.
Some form of Vulgar Latin was then the ancestor to the Romance languages.
The Romance languages include Spanish, French, Italian, Romanian, etc.

The most common, and not too critical, error I see in descriptions is calling Romance a daughter or sub-family of Proto-Indo-European directly. That should technically be Italic, with Romance another level or two below that. Whether "Latin" is the same as Proto-Romance depends on what you mean by the question and by "Latin".

Well, I found this http://rationalwiki.org/wiki/Le_fran%C3%A7ais_ne_vient_pas_du_latin 

I would be really surprised that SCIENCE magazine, even in a news article would stoop to giving credence to this kind of book, but I cannot find anything else.

I was reading in some publication, I think it was PLOS, but regrettably cannot find the link, that there was some debate as to whether Latin was the parent language of the Romance languages.

A language is called "romance" or "romanic", if it is basically like, or descends from, Latin. So, in principle, a given language may not be both "romance" and "Unlike Latin". However, since Latin is demonstrably descending from Greek, it is quite possible for a language to be both "romance" and "not descending from Latin". In that case, the cognate words of this language and of Latin have Greek as their common ancestor. (Many years ago, I started reading the poems of the Troubadours, which are in Occitan or Provenzal, and I was able to understand them by automatically utilizing the cognates in this language and my knowledge of Italian and of Latin. One who knows only Greek should be able to do the same. Only a few words here and there were incomprehensible -- probably of Franconian origin. The same happened to me again when I was reading some literature written in the Lombard dialect of Italy, while my native dialect in southern Italy is quite different and incomprehensible to the Lombards.)