On differing names:
Radicals and it's names is a wide subject, and such this topics require to have in mind that grammar and its nomenclature evolves through time (as the language itself). In linguistics, this kind of issue pops up all the time, and to explain the variations we have to look back at the language history (this approach is called diachronic).
With that in mind, let's say that some times some scholars seems to prefer a specific nomenclature and classification pattern, and sometimes they prefer others. Once your objective is the Kanken test, we do not need to explain the variations so let's take a look of how they behave in modern japanese.(this approach is called synchronic).
On the radicals names:
You seemed to ask generally about the names of all radicals, but because there are plenty, I'll just treat the examples you've highlighted. The 乙、乚 and the 丨
乙 is called おつ。おつにょう, as you've suspected, refers to 乙繞, 繞 being the radicals written from the left to the bottom. (like 辶) 乚 is the same radical when written in the right (like a 旁「つくり」, but it'is not), but once this radical comes the representation of "something that obstructs the growing of other something", traditionally it as been read just as another form of 乙, in fact they have the same name.
Both of them are おつ, but maybe that's why when referring to 乙、usually、one would say おつにょう。
About 乙部 「おつぶ」this is not the traditional name of the radical, indeed it is not hard to find it, but it's as an abbreviation of 乙繞部首 "the radical otsu"
As the Grammatical terms usually require kanji with lots of strokes, it is common to abbreviate the nomenclature (e.g. like the informal classical verbs conjugation: サ行変格活用ー＞サ変)
In fact we could generalize that no radical has 部 in the end of it's name. Whenever we find it, we must read it as an abbreviation of the grammatical nomenclature.
The same goes for 丨 the name can be either ぼう or たてぼう.
But as your aim is the Kanken test, (maybe) you should not worry too much about it (for taking the test), because just as @Ben's answer well shows, the kanken test do not demand you to write the radical name.
Another point is that maybe you might not study all radicals starting from the ones with less strokes, FOR THE PURPOSE OF THE TEST. Once every level of Kanken has a predefined list of kanji (that goes with the japanese scholar years, level 6 corresponding to 5th grade) maybe sticking to this list could be way more effective.
For the Kanken test, I strongly recommend you the preparatory book series published by the very Kanken Association. ()
They have a book for every level.
I have answered based on my personal experience with the help of the kanji practice manual 漢字学習ステップ published by 日本漢字能力検定協会、the japanese kanji dictionary 例解学習漢字辞典 published by 小学館、and by the japanese language dictionary 広辞苑 published by 岩波書店。