Are the $\lambda_I$-Calculus and the $\lambda_K$-Calculus equivalent?
You've basically answered the question yourself.
$\lambda K$ is just another name for the standard, untyped lambda calculus.
$\lambda I$ is a strict subset of $\lambda K$.
$\lambda I$ doesn't allow terms where one abstracts over a variable but doesn't use it. So
$$K = \lambda xy.x \in \lambda K$$
but
$$ K \not\in \lambda I$$
Thanks to this restriction, $\lambda I$ has some interesting properties, in particular if $M$ has a normal form then so do all its sub-terms.
Barendregt, H. P. The Lambda Calculus: Its Syntax and Semantics contains some notes about $\lambda I$, namely:

... the $\lambda I$ calculus is sufficient to define all recursive functions  (since $K_1 := \lambda xy . yIIx$ satisfies $K_1xc_n = x$ for each of Church's numerals $c_n := \lambda fz . f^n z$ - it is also the case that for each finite set $n$ of nf's, we can find a &quot;local $K$ for $n$&quot; $K_n$ such that $K_nMN = M$ for each $N$ in $n$).
... The $\lambda I$ calculus corresponds to the combinatory logic with primitive combinators $I$, $B$, $C$, and $S$. ...