I know it depends on who you ask, or what book you are reading. So for the purpose of this post I am going to define "most likely meaning" to be the meaning that would be attached to a word or phrase by a clear majority of working mathematicians. To avoid making this too complicated I will leave "clear majority" and "working mathematician" as primitive notions...
I have been wondering about the difference between the most likely meaning of "irreducible algebraic set" and that of "variety". A web search and of the archives of Stackexchange and of various books was inconclusive.
So suppose I give to the phrase "irreducible algebraic set" the meaning: an algebraic set $V$ over some field $k$ such that if $V= V_1\cup V_2$ for some algebraic sets $V_1$ and $V_2$, then either $V_1=V$ or $V_2=V$. Equivalently, this means that the corresponding ideal in $k[x_1,\ldots,x_n]$ is prime. Then I take the meaning to the word "variety" to be: an irreducible algebraic set over some algebraically closed field $k$.
Are those the most likely meanings of "irreducible algebraic set" and "variety"? If not, how do we distinguish a variety from an irreducible algebraic set?
The notion "variety" is very bad in the sense that it has non-equivalent definitons. There are many more than the ones you stated. When I read the word variety I directly check how the author defines that. In the classical sense (not using scheme theory) I would say that irreducible algebraic sets or just algebraic sets over $k = \overline{k}$ are the most common ones. When using schemes, the ones I have seen the most are separated $k$-schemes of finite type, integral separated $k$-schemes of finite type, geometrically irreducible separated $k$-schemes of finite type and reduced separated $k$-schemes of finite type.