They are defined by their position relative to their own political systems. Those who defends the status quo, or wishes to roll back to a real or imagined status quote ante are considered right wing. Those who supports change in the other direction are then considered left wing. This classification originated in the Estates General during the French Revolution: those who supported the king, hence the status quo, were seated on the right; those who supported the revolution, the left.
Beyond this simple contextual divide, there is no absolutist, universal political measure. This is also why the left-right spectrum is generally not applied to pre-Revolution figures. Which is also why assigning "liberal" or "conservative" labels to historical figures usually is a complete hogwash.
One exception in your list is Cato the Elder. Cato was a conservative because he believed in preserving antiquated Roman views. This is a rare non-ambiguous case of being conservative in the literal sense of the word.
As for the rest, Thomas Jefferson was a "liberal" only to the extent that Hamilton was a "conservative". And Hamilton is only regarded as a conservative on the basis that he advocated for institutions, strong central authority etc, i.e. what was then the political mainstream.
Hitler was right wing, but less "conservative" and more of a reactionary because he wanted to "restore" Germany to a glorified past. Stalin, likewise, was a left wing dictator, but not really "liberal". Communism was simply regarded as left wing because it challenged the prevailing capitalist systems of the world. Ultimately, there is scant little difference in substance between the two brutally authoritarian tyrants.
Not all Republicans were "extreme left" during the American Civil War. The Radical Republicans were a specific faction considered far left "radicals" because they wanted to abolish the institution of slavery posthaste. This was defined in contrast to the moderate Republicans led by Abraham Lincoln. The latter were regarded as more conservative because they were willing to be more muted on slavery or even tolerate it in order to save the union.