To start off, Siberia got "lexis" and "lexicon" switched, so that may be where some confusion is coming from.
As Kearsley says, a lexicon is a language's inventory of lexemes. A lexeme is an abstract concept, the unit of meaning underlying various inflected forms. For example, in English, we have this set of words, all connected through inflection: sing, sings, sang, sung, singing. They are all understood as forms of a verb meaning "produce melodic sound", but the inflections add grammatical information such as tense, aspect, person, number and finiteness. The lexeme is that underlying verb, without inflection. The lexeme of the verb forms I gave would generally be represented as "sing".
A lexis is all the actual words in a language, including inflections; as opposed to abstract lexemes. As a result, a lexis of English would include "sing", "sings", "sang", "sung" and "singing", not just "sing".
As for vocabulary, Kearsley's definition feels quite straightforward to me; the term is about a group of words in a particular context. I'll add that "vocabulary" is sometimes a synonym for "lexicon". For example, "English has a large vocabulary" normally means the same thing as "English has a large lexicon".
As Kearsley points out, there's a lot of discussion about what, precisely, constitutes a "lexeme" or a "word". I'm no theoretician, and I don't have the knowledge to open that particular can of worms.