Which Machine out of Induction machine, separately excited DC machine and permanent magnet DC machine will have the highest Torque to weight ratio?
In a  motor (and in fact in many other situations, power supplies for example), some things are controlled by the physics, but for other things there is more of a free choice dictated by economics, or even the market fashion.

In a motor, power density is very closely controlled by physics. The power to weight is limited by the amount of copper, iron and magnetic material in the motor. Here the permanent magnet DC motor wins, especially strong-field brushless types now available.

Power is the product of speed and torque, and for any given motor, there is more of a choice between them. Not an unconstrained choice, but there is some freedom within the power limit. Altering the number of copper turns alters the ratio of torque and speed the motor produces. Some motors, for instance vacuum cleaners, are optimised for very high speed, and have very mechanically strong rotors and few turns. Those intended for large-prop quad-copters have high torque and relatively low speed, certainly not challenging the rotor strength.

So your highest torque motor is likely to be found in the PMBLDC catalogues, and it's likely to one of the bigger one your dollar budget can afford. Which of those motors though requires you to read all the specs and compare.