The person is perfectly correct, except with the last sentence.
I guess this is a semantical problem. You can say "X branched off from/of Y", which says X is a descendant of Y, or "X branched away from Y", which just means they share roots and developed in different directions. Steve said the latter (common root: classical Kodokan Judo, branches: BJJ and modern Judo), you read the former (modern Judo branching off BJJ) in the first sentence. That is not written there, though. Steve even specified that BJJ branched off of Judo after that. Also, you seem to have no idea what the actual diversity of techniques or histories of the two is, they are both quite old and consist of much more than you make of them, actually.
Also, Europeans generally refer to BJJ and Jiu-Jitsu/Ju Jutsu as distinct arts, mostly meaning JJJ or descendents (like the German Ju Jutsu self-defense system) when saying the latter. The equalization of Jiu-Jitsu with BJJ is a rather recent and generational development prevalent mostly in the Americas due to the more widespread popularization of BJJ compared to JJJ there IMHO. With regards to spelling, Steve was only being picky there since ju jutsu and jiu jitsu are the same thing with different romanization. Thus, in that point, I'd say the call for specification is justified, the call for distinguishing by spelling is not.
Kodokan Judo as the martial art BJJ evolved from
The training of Carlos Gracie under Mitsuyo Maeda (7th dan Kodokan Judo) started in 1917. This is three years before the last major overhaul of Kodokan judo (systematization of Gokyo and several katas) under the supervision of Jigoro Kano himself and 13 years after Maeda left the Kodokan to go abroad, long before Judo and other Japanese martial arts were intentionally tamed and branched after WW II.
Yes, it is true that Jiu-Jitsu (Japanese Ryu) and Judo were not clearly separated terminologically back then. Actually not before 1925 even in Japan. But it is clear that Maeda was a Judoka - and not a Jiu-Jitsuka. He was one of the most renowned second-generation students and teachers of the Kodokan. He inspired people like Mifune (later 10th Dan and generally considered one of the all-time greats of Judo) while training and teaching there. He never in his life trained a Jiu-Jitsu Ryu of any kind (only Sumo) before or after joining the Kodokan. He founded a Judo academy in Brazil where he taught Kodokan Judo and was awarded a Kodokan Judo dan rank one day before his death in 1941. Thus, while Jiu-Jitsu as a term for Kodokan Judo was common back then, what the founders of BJJ learned was exactly what is today known as: Kodokan Judo. Only because Judo was not used regularly to refer to the art, it is called Brazilian Jiu Jitsu and not Brazilian Judo. What the Gracies (just like Kodokan students) learned was much more open and fight-oriented in its applications back then, though. So saying that BJJ branched off of Judo, while Judo branched off of Japanese Jiu Jitsu Ryu (even if the terms were not coined this way back then) is perfectly valid.
Why is modern (Olympic) Judo so different from modern (sports) BJJ?
The confusion may arise due to Maeda's own take on Judo, which was actually common among first- and second-generation students of the Kodokan: He joined Judo because of reports of Judoka winning no holds barred matches against Jiu-Jitsu schools. His later sensei won against a fighter who was considered one of, if not the, strongest Japanese fighters of the time. Thus, Maeda considered all possible applications of Judo, in all ranges, as means to win a fight, no matter the ruleset, whether you fight with a jacket or without, how big your opponents are, or which martial arts background they have. He continuously challenged and fought fighters of various martial arts, including armed opponents, all around the world and further refined applications.
That Maeda's Judo revolved around a lot more groundwork than contemporary Olympic Judo is probably also due to him being relatively small and light for a fighter challenging people regardless of weight and style, ie. him playing his (technical) strengths and averting his (physical) weaknesses. But he won a lot of his fights by throwing his opponents and was feared (among students no less) for the power of his throws. In no way it is valid to say that BJJ only took the groundwork. Maeda practiced, applied, and taught the whole variety of grappling applications in all phases of combat (ranged, clinch, ground, even weapons).
(Kodokan) Judo (still) involves all aspects of fighting, including strikes and kicks, weapons, self-defense, and, of course, ground fighting (which is part of Olympic Judo as well, as beautifully exemplified in this video posted by Dave Liepmann in a comment). It is just not trained very often as it is not (an important) part of Olympic Judo as it evolved since its inclusion into the Olympic curriculum in 1964. There even was a change in rules which aimed at encouraging more groundwork in competition again recently.
As a matter of fact, we now have a similar development of narrowness within BJJ itself, where sports BJJ starts to focus solely on pulling guard and groundwork, completely dismissing the original MMA and self-defense perspectives the Gracies inherited from Maeda. BJJ was much more multi-faceted and still includes throws. Gracie Jiu-Jitsu is still much wider with self-defense and all aspects of fighting trained. But this development towards groundwork "only" within BJJ is actually very recent, more like not before the late 90s. I think the reason why sports BJJ so excessively concentrates on groundwork is that it is the furthest away from striking distance, in which the founders of the style and their teacher naturally felt least comfortable. The famous early Gracie MMA fights, therefore, took place on the ground after distance was covered: To be in the phase of combat strikers - their most common opponents - feel least comfortable. Thus, groundwork became linked to BJJ and vice versa. It is mistaken to think that it has always been that way, though. And there are voices which call for a change of rules to make throws more versatile in competition. But truth be told, I expect that the groundwork focus remains exactly in order to make it a sport distinct from Judo beyond "we also do no-gi".
Intersections between BJJ and Judo
We do have a lot of mutual exchange between BJJ and Judo. Just think of Travis Stevens, who was awarded BJJ purple belt after a few lessons due to his world-class Judo expertise. Grappling is grappling is grappling, so there are quite a few BJJ people training Judo to up their standing game, as well as Judo people training BJJ to up their ground game. Both is part of both sports branches, but both BJJ and Judo (originally) are more than sports and very similar in their technical philosophy of efficiency. Therefore, both techniques, and strategies are repeatedly going both ways.
Traditional Brazilian Jiu-Jitsu is, in fact, a descendant of Kodokan Judo. Nobody ever stated anything other than that. It has evolved differently as it focused on applicability in all possible fighting situations - with and without a jacket - while Olympic Judo started to focus on throws and very few ground techniques as direct follow-ups when going to the ground without an ippon, fighting in jackets only. They both share the focus on grappling and the omission of striking in order to be able to use full-intensity randori in training. They both are based on the technical use of body mechanics, efficiency, and positioning. Both classical Gracie BJJ and Kodokan Judo still are more multi-faceted than their sports counterparts and are, as a matter of fact, very similar up to this day, all things considered. Actually, the most extreme differences of technical content besides training focus are a) BJJ training without gi and b) Judo training kata, as far as I am aware.
Judo was just internationalized much earlier (start of 20th century) and is decades ahead with regards to diluted expertise in all aspects of the art, may it be to it spreading fast or being a codified sport. As a matter of fact, many Judo federations have projects to encourage self-defense and groundwork to be integrated into the training more regularly, including above-mentioned changes of competition rules. BJJ is still in the process of developing as a sport and there already are intentions to establish awareness of the breadth of the original BJJ and train more diversely.
Generally, you seem to have a biased view based on the very "sportified" aspects of the arts you get to see in competitions and are propagated as simplified labelling of the arts: Judo - throws, BJJ - groundwork, Muay Thai - clinch/knees/elbows, etc. It is where they excel in average, not all they are able of.
Asking to use "jiu jitsu" for BJJ only is unwarranted in any case, though.
(Sources can be found in this surprisingly good and well-sourced Wikipedia article on Maeda)