The source for this is a Gemara in Berachos, 39a:
אמר רב פפא פשיטא לי מיא דסלקא כסלקא ומיא דלפתא כלפתא ומיא דכולהו שלקי ככולהו שלקי
Rav Pappa said "It is evident that beet-liquid is like a beet, radish liquid is like a radish, and liquid of all boiled (produce) is like the produce"
There are several interpretations of the Rishonim as to the reason and parameters of this halacha, particularly in contrasting it with an earlier Gemara (38a):
ואמר מר בר רב אשי האי דובשא דתמרי מברכין עלויה שהכל נהיה בדברו מ"ט זיעה בעלמא הוא
And Mar son of Rav Ashi said "The blessing on date honey is shehakol. Why? Because it is just the sweat [of the fruit]."
The resolution of the Rishonim has ramifications for the absolute truth of both your first and second statements.
The Rosh (Siman 18) seems to understand that it is based on the transference of taste in the juice (I am over-simplifying, but don't want to write a dissertation here), and seems to suggest that the difference between the two is squeezing versus boiling. He concludes (as a possibility which is accepted in many poskim) that boiling fruit would therefore have the same halacha. The Tur and R' Yechiel (Tur O.C. Siman 205 towards the end) had a brotherly dispute if squeezing vegetables was the same as squeezing fruit or cooking vegetables, with the Tur assuming they are like squeezed fruit, and R' Yechiel held they are at least as good as cooked vegetables.
The Rashba (Chiddushim to Berachos 38a) holds there is no qualitative difference between fruits and vegetables, nor between cooking and squeezing, and it all depends on the quality/significance of the juice, which is determined by normalcy to squeeze/cook the produce in question. Therefore, some fruits would get a blessing of ha'etz on the juice, and some vegetables would get a shehakol on the juice.
The Ritva (Chiddushim to Berachos 39a) actually holds that your entire claim is false. He understands the Gemara to mean that if you eat vegetables together with their juice then the blessing on the vegetable exempts the juice. But the juice on its own would get a shehakol.
The Ran (s.v. ומיא דשבתא), in one answer, seems to understand that vegetables are still considered vegetables even in juice form because they are still eaten like vegetables, namely you dip bread into them and eat them like a solid, not like a liquid. It seems to follow that a vegetable juice drink would not retain the status of the vegetable.
In conclusion, some vegetables retain vegetable status even in juice form, according to most Rishonim.
The Shulchan Aruch concludes that if you squeeze vegetables, you make a shehakol (O.C. 205:3):
אם סחטן אינו מברך על אותם משקין אלא שהכל
and that all squeezed fruit juice except wine and olive oil gets shehakol (not like the Rashba) (O.C. 202:8):
דבש הזב מהתמרים מברך עליו שהכל וכן על משקין היוצאין מכל מיני פירות חוץ מזיתים וענבים מברך שהכל
and he cites the opinion of the Rashba (not by name), followed by the Rosh, regarding cooked fruit (202:10):
פירות ששראן או בשלן במים אף על פי שנכנס טעם הפרי במים אינו מברך על אותם המים אלא שהכל והרא"ש כתב דאפשר שאם נכנס טעם הפרי במים מברך בורא פרי העץ
which seems to imply that he concludes like the Rashba, and juice from cooked fruit is shehakol.
Regarding the seeming contradiction in the Shulchan Aruch from 202:8 to 202:10, there are various suggestions in the commentaries, although my personal opinion is that the Shulchan Aruch is not concluding like the Rashba, but rather like the Rambam (Berachos 8:2&4), who seems to hold without qualification that fruit juice is shehakol, and the distinction of the Gemaras is fruits vs. vegetables.