I think it derives from the word всё witch means everything, or all of, and is a very all-encompassing word, as is совсем. The prefix со- sometimes used to make a verb perfect form (if now that is the right term), like творить, сотворить. Adding co- implies your will be done with it. Совсем therefore means "completely" or "totally".
It depends on what совсем and не act upon, which is affected by word order in the most predictable way:
совсем новый = completely new
- совсем не новый = completely (not new) = not new at all
- не совсем новый = not (completely new) = not quite new
совсем забыл = completely forgot
- совсем не забыл = totally (did not forget) = did not forget in the slightest
- не совсем забыл = did not (forget completely) or not exactly "forgot"
We do not have that many exercises with совсем (which is a pity because it is a very popular word). I looked through them all, and could not find the one you are talking about.
Совсем means "totally", "at all". By some leap of logic you may say that "definitely" is, in some sense, in some context, once in a blue moon approximately the same as "at all". However, this will only apply to contexts where such substitution makes sense.
This way -ский adjectives can produce adverbs that express the "manner of action". It is exactly what говорить attaches:
- английский → по-английски
- русский → по-русски
- польский → по-польски
- французский → по-французски
- японский → по-японски
- финский → по-фински
It is a useful structure to remember: a lot of popular languages/countries are associated with an adjective ending in -ский.
Its use is different with verbs, since совсем will imply an action performed to an extreme extent. "Understanding very much" is not well defined in Russian because you either have understanding or lack it to some degree.
However, совсем is very widespread in negative sentences as совсем не ("not ... at all"). You can also encounter совсем with verbs that mean a loss of some quality or generally carry a negative connotation (e.g. "Я совсем устал", "Она совсем забыла про нас").
For other verbs it is generally a good idea to look for corpus examples: we don't use совсем with that many verbs in positive sentences.
I found a great reply from user zirkul on this discussion board. This user wrote:
There is a slight difference between "Вера говорит по-русски" and "Вера говорит на русском". To me, "Вера говорит по-русски" could mean either Vera's general ability to speak Russian or the fact that is it talking to someone in Russian (you've overheard her and ask what language she is speaking). "Вера говорит на русском" is only applicable to the latter.
So what I take from that is:
говорит по-русски = speaks Russian
говорит на русском = speaks in Russian