"Home of Record" is used to determine travel entitlements when one separates from the military. It has nothing to do with voting or paying taxes, registering vehicles, nor any of the other priviledges of state residency. "Home of Record" can only be changed if there is a break in service of more than one day, or to correct an error.
"Legal Residency," or "domicile", on the other hand refers to the place where a military member intends to return to and live after discharge or retirement, and which they consider their "permanent home." Legal residency determines what local (state) tax laws a military member is subject to, and in which local (city, county, state) elections they may vote in.
Does that mean a military member can change their "legal residence" anytime they want, and therefore avoid paying state taxes? Not quite. Under the law, "legal residence" is the place that the military member intends to live after they separate or retire from the military. It's the place that they consider their "permanent home."
However, the military is required by regulation to ensure that military members are not changing their "legal residence" for the sole purpose of obtaining a tax advantage. Therefore, when changing your "legal residence," military officials at the legal office (or finance office) may require some degree of proof that you consider the new state to be your "permanent home."