Super Saiyan Grades II and III
These are the two extremely muscular forms of Super Saiyan which Vegeta and Trunks use when fighting Cell, and which Goku briefly displays while training in the Room of Spirit and Time. These are ultimately established to be less than ideal ways of surpassing Super Saiyan, and are not as well-balanced as the regular Super Saiyan form.
Manga and Anime
In Chapter 378, Vegeta uses his new buff Super Saiyan form to completely outclass Cell, and calls himself “Super Vegeta” (which appropriately enough is the actual chapter title). The “Super” here is the kanji 超 (usually read as chou when used as a suffix) with furigana indicating it should be read as the English word “Super” (スーパー/suupaa). This is the same way the “Super” in “Super Saiyan” is typically written. It’s somewhat of a generic term, and is also used in the manga to refer to Super Saiyan 3 Gotenks and Super Saiyan Vegetto, so it’s not tied specifically to Vegeta’s form when fighting Cell. Some of the earlier video games like Legends or Final Bout also used 超 as a prefix for any Super Saiyan character (so regular Super Saiyan Goku is just “Super Goku”), though it’s not clear in this case whether the kanji’s supposed to be read as chou or suupaa. The anime narration does the same thing too sometimes, especially with characters Super Saiyan 2 and above, so that Super Saiyan 3 Gotenks is called “Super Gotenks 3” and Super Saiyan 4 Goku is “Super Goku 4”. Anyway, regardless of all that, the phrase “Super Vegeta” is probably the one most strongly associated with this form, since Vegeta uses it himself in the actual story.
So Vegeta’s got his super-muscular form, which he calls “Super Vegeta”, while Trunks has the same form plus an even more muscular one (which is unfortunately slow as hell). Confusingly, this even more muscular form is called “Super Trunks” in chapter 386’s title (“Super Trunks Has Surpassed his Father”), as well as the title of anime episode 165 (“Super Trunks Has a Weakness!! Cell’s Shocking Bombshell Declaration”). This is the same “Super” written 超 but pronounced like the English word.
Then in chapter 387, Goku shows off these same two forms to Gohan in the Room of Spirit and Time. First he reaches Super Vegeta’s form, then says that “1 further grade of transformation is possible” (さらにもう一段階の変身が可能だ), at which point he reaches Trunks’ form. 段階/dankai here can mean things like grade, level, tier, stage, etc, and it’s this line that the names “Grade II” and “Grade III” for these forms comes from, but more on that later. Viz translates the line as “and I can do one more notch!”, so going by that you could call them “Notch II” and “Notch III”. Also, when Trunks is waiting for the right time to transform into his “Grade III” form, Kuririn finally tells him that now’s the time to become that “outrageous Super Saiyan” (とんでもねえ超サイヤ人/tondemonee Suupaa Saiya-jin). Viz changes this to “Super, Super Saiyan”, which might be a take-off of the common fan name for the form, “Ultra Super Saiyan” (more on this later). “Ultra” means the same basic thing as “Super”, so Viz calling the form “Super, Super Saiyan” might be a joke on the fan name.
Guidebooks
Going off of the Goku line mentioned above, Daizenshuu 2’s Growing Up section names Vegeta and Trunks’ muscular form “Super Saiyan Grade II” (超サイヤ人第二段階/Suupaa Saiya-jin Dai-ni Dankai) and Trunks’ ridiculously muscular form “Super Saiyan Grade III” (超サイヤ人第三段階/Suupaa Saiya-jin Dai-san Dankai). Remember that during the scene in question Goku starts out a regular Super Saiyan, transforms into the muscular form, says another “grade” of transformation is possible, then transforms into the really muscular form. So if regular Super Saiyan is counted as the first grade, that would leave the two muscular forms as a second and third grade. This gets confusing because of course in the Boo story arc Goku introduces the terms “Super Saiyan 2” and “Super Saiyan 3” for a pair of completely different forms. The main difference between these two groups of names is that the muscular forms are named according to Japanese numbering (第二段階/Dai-ni Dankai and 第三段階/Dai-san Dankai) while Super Saiyan 2 and 3 are read as the English numbers (超サイヤ人2/Suupaa Saiya-jin Tsuu and 超サイヤ人3/Suupaa Saiya-jin Surii). That’s why I personally like to use Roman numerals when writing about the Grade forms, just to visually distinguish them from Super Saiyan 2 and 3 as much as possible.
The third TV Animation daizenshuu has a chart of the various Super Saiyan forms, and it uses the same names for the grade forms as Daizenshuu 2, but also groups them together with Super Saiyan Full-Power and labels them all as “Variations” (using the English word) of Super Saiyan 1. The GT Perfect Files Vol.1’s section on Super Saiyans likewise uses the same names as Daizenshuu 2.
Video Games
Though they’re not playable characters, Legendary Super Warriors has two cards corresponding to the Grade forms, which it calls “Super Saiyan Grade II” and “Super Saiyan Grade III” like in Daizenshuu 2. The English release of the game though mistranslates these as “Super Saiyan 2” and “Super Saiyan 3”. Outside of that though, the “Grade” naming scheme hasn’t been used in games, in large part probably because the forms themselves don’t appear too often. The Sparking!/Budokai Tenkaichi and Raging Blast games have them, but calls them “Super Vegeta” and “Super Trunks” like in the series. “Super Trunks” still refers to Grade III; these games typically split sword-less Trunks off as a separate character, with Grade II just being treated as its default Super Saiyan form, which then transforms into Grade III/Super Trunks.
Fan Terms
The earliest fan term for these forms (in the English-speaking fandom, at least) goes back to Curtis Hoffman’s manga summaries. These were some of the earliest in-depth information available on the series in English, and while in retrospect we can see that they really kinda sucked, at the time they served almost like a Bible for many fans. In the introduction to volume 32’s summary (where he recaps events of the previous volumes), Hoffman writes:
And with that fateful “I guess”, one of the most persistent fan terms was born. “Ultra Super Saiyan” has remained a very common name for the Grade forms, to the point where former Viz editor Jason Thompson matter-of-factly uses the term during his “House of 1,000 Manga” write-up of DB. Now, we should first distinguish between the way Hoffman himself used the term, and the way later fans have. First off, despite using “Ultra Super Saiya-jin” in the quote above, Hoffman seems to have at all other times used simply “Ultra Saiya-jin”. In fact, he talks about “Ultra Saiya-jin” Vegeta and Trunks in his intro to his volume 31 summary, a volume before “Ultra Super Saiya-jin” turns up, but that intro draws on information from later volumes, so it probably was written later than his volume 32 summary for whatever reason (in fact, it’s not really clear why he talks about those forms at all for volume 31, since they don’t debut until 32).With volume 31, comes the introduction of the idea that the heroes can go beyond Super Saiya-jin to become something else. (Ultra Super Saiya-jin, I guess.)
The second thing to note is that Hoffman used “Ultra Saiya-jin” as a catch-all term for the various different upgrades on Super Saiyan that appear before the introduction of Super Saiyan 2. So he calls Full-Power Super Saiyan Goku and Gohan “Ultra Saiya-jin” as well, and doesn’t distinguish between Grade II or Grade III at all (they’re both just “Ultra Saiya-jin”). Many later fans though have taken to calling Grade II “Ultra Super Saiyan” and Grade III “Ultra Super Saiyan 2”. On the other hand, some fans call Grade II “Ascended Super Saiyan” and reserve “Ultra Super Saiyan” for Grade III. The phrase “Ascended Super Saiyan” apparently comes from the Funi dub, which uses “ascend” as a slightly fancy translation of koeru/“to surpass”, which gets tossed around a whole lot in the series when talking about stronger kinds of Super Saiyan. While I hear the dub flat-out calls Grade II “Ascended Super Saiyan”, I’m also pretty sure they use the term to refer to Super Saiyan 2 as well, and call Super Saiyan 3 “double ascended” or something like that. So “Ascended” apparently isn’t tied to any specific kind of Super Saiyan, but gets used for many different forms, the same way “Super” is in Japanese.
A lot of fans have also mistaken Trunks’ Grade III form as actually being Super Saiyan 2. For instance, while researching all this I found a second old summary of volume 33 which was also supposed to be by Hoffman. I think it probably isn’t, but either way, it calls Grade III “Super Saiya-jin 2 (wrong way)”. The DBZ/Budokai game series gave Trunks the Super Saiyan 2 transformation for some reason, which probably contributed to the idea.
Well that's all for now. If you're not confused by now, I probably haven't done my job.
I learned so much from your various guides!