Given the question:
I am not sure what $Ha=Hb$ actually means.
I can read it as:
$ah=bh$ for all $h\ in H$
Or $ah_1=bh_2$ for all $h_1,h_2 \in H$
Or for all $h_1$ there is a $h_2$ such that $ah_1=bh_2$
And all of these readings are different mathematically. What is the correct reading of that notation?
$Ha$ is defined as $$Ha := \{ha \mid h \in H\},$$ so $Ha=Hb$ refers to an equality between two sets.
It turns out that for every $a,b\in G$ and every subset $H$ of $G$, the sets $Ha$ and $Hb$ are either equal or disjoint. That means that $Ha=Hb$ is equivalent to $Ha \subseteq Hb$, which in turn is equivalent to "for every $h_1 \in H$ there is some $h_2 \in H$ such that $h_1 a = h_2 b$".
$aH$ and $bH$ are defined analogously, and are always either equal or disjoint. So your third interpretation is equivalent to them being equal.