Determining equivalence classes of $\{w \in \{0,1\}^*\mid$ the $k$-bit of $w$ from the right is 1$\}$
http://users.utu.fi/jkari/automata/fullnotes.pdf explains equivalence classes well (under section 2.8 "Myhill-Nerode theorem") whence the following notation and definition is borrowed.

Define $\text{Ext}(w,L) =\{x\mid wx\in L\}$ where $w$ is in $\Sigma^*$ and $L$ is the language, ie. the set contains of all suffixes that, when added to $w$, yields a word in $L$.

Two words $a$ and $b$ are in the same equivalence class if $\text{Ext}(a,L)=\text{Ext}(b,L)$.

For the language $L_k$ described, only the last $k$ letters of $w$ matter when figuring out $\text{Ext}(w,L_k)$ for any word $w$ because the other letters do not matter as they do not contribute to the condition "the $k$-th bit of $w$ from the right is $1$". Since there are only finitely many combinations for the last $k$ letters, specifically $2^k$, there are a finite number of equivalence classes in $L_k$.

As per your edit about suffixes of length less than $k$, sure, $\text{Ext}$ can contain such suffixes, but they do not matter in showing that there exist finitely many equivalence classes.