By definition, lynching is a form of murder via hanging.
I, and I think most people, don't consider capital punishment via hanging as a lynching. Two different animals.
Was curious as to who decides the form of punishment for treason
Punishment of Treason
The Congress shall have Power to declare the Punishment of Treason, but no Attainder of Treason shall work Corruption of Blood, or Forfeiture except during the Life of the Person attainted.
ARTICLE III, SECTION 3, CLAUSE 2
Under common law, punishment for treason generally included DRAWING, HANGING, BEHEADING, and QUARTERING. As with other crimes carrying sentence of death, those adjudged guilty of treason and finally sentenced were considered attaint, or stained, meaning dead in the eyes of the law—even before execution. Once attainder was established, the attainted forfeited his real estate to the Crown—a requirement symbolizing lack of entitlement to the benefits of society. Attainder also worked corruption of blood, preventing the attainted from inheriting or transmitting property and preventing any person from deriving title through the attainted. Forfeitures and corruption of blood worked hardship on dependents and relatives in order to provide maximum deterrence. Eventually, Parliament modified the laws of forfeiture and corruption of blood to protect the innocent.
According to the Constitution, punishment can be set by Congress, but not to include corruption of blood or forfeiture extending beyond the offender's life. Quite apart from this limitation, Justice Joseph Story notes the explicit grant of congressional power over punishment was intended as a leniency, to preclude the assumption of the common-law punishment's harshest elements. The First Congress used its constitutional power of declaring the punishment for treason by establishing the penalty of death, with seven years' imprisonment for misprision of treason.
The actual punishments for those convicted of the federal crime of treason have generally been more lenient than the statutory maximums. Those convicted for their part in the Whiskey Rebellion were pardoned by President George Washington. The United States government regarded Confederate activity as a levying of war, but all Confederates were pardoned by presidential amnesty. Max Haupt, convicted for giving aid and comfort to his alien son, was spared death and sentenced to life imprisonment. (His son Herbert was convicted by a military tribunal for his role as saboteur, and executed.) Tomoyo Kawakita, convicted of treason for abusing American prisoners of war, was sentenced to death but had his sentence commuted to life imprisonment by President Dwight D. Eisenhower. By contrast, the Rosenbergs' espionage convictions brought death sentences.
Of the two successful prosecutions for treason at the state level—Thomas Dorr in Rhode Island in 1844 and John Brown in Virginia in 1859—only Brown was executed. Dorr was pardoned, and elements of the political agitations for which he was convicted were soon adopted into law in Rhode Island.
https://www.heritage.org/constitution/articles/3/essays/120/punishment-of-treason