Choose by measured payback
A more expensive axe is only a better choice when the extra wood it produces repays the cash cost within a useful amount of play or away time.
The official Roblox description confirms the core role: chop trees to get wood, upgrade your farm, sell wood for cash, and let axes farm offline. It does not provide an axe catalogue or formula. That means a named S-tier ranking would look confident without having the evidence to support it.