Actuation is a complex subject that cannot be reduces to 1 component alone. It is the interaction of software electronics and mechanics. "Toy" servos tend to incorporate everything in one small package. Industrial robots have 3 or 4 separate components (each costing several hundred or a few thousand euros) instead.
To answer your question, worm gears are almost never used in the joints of industrial robots. Mostly cycloidal or planetary gears are used.
Transmission:
Many hobbyist project use small "toy" servos which are, basically a DC motor, some simple position feedback system and a transmission (+ the controller). The transmission is a 1, 2 or sometimes 3 stage simple gear reducer. Due to its low quality it sometimes has large backlash, so the output of the transmission can move a bit (approx. 0.1 degrees, value varies on quality) even if the motor does not move. This leads to uncontrollable motions, vibrations due to jerk, etc.
The best "toy" servos you can get, to my knowledge are the Dynamixel servos.
Industrial robots us professional gearboxes with backlash below 0.2 arc minutes. Price for something that you pointed out in the picture is probably around 1000-3000 EUR. Typical manufacturers: ZF, Wittenstein, Nabtesco, Bonfiglioli.
Furthermore, the control algorithm applied backlash compensation to compensate these and further improve the precision.
Motor:
As mentioned, the "toy" servos use DC motor in most cases. DC motor in general my not be cheaper then PMSM (what industrial robots use, see below) but due to bulk manufacturing in small sizes and cheaper motor drivers, overall in the small size range, they may be cheaper to use.
Industrial robots use Permanent Magnet Synchronous Motors. They are less expensive in general as DC Motors (also due to lower maintenance costs, since there is no brush to change regularly). Power ranges for smaller robots are a few hundred watts for larger Robots a few kilowatts. Price for such a motor varies from approx. 300 EUR to 2000 EUR for the high powered ones. Most of them include (optionally) resolver or encoders for position control. These have high precision (up to 4048 values per rotation with additional signal for in-between values). Typical manufacturers form for motors ABB, Siemens, Bosch, Parker. For best performance, encoders are integrated in the motors.
Motor Controller
Toy servos use in most cases 1 Position control loop with PID control method. This is very simplistic and delivers bad performance.
Industrial motor controllers use cascaded control loops (each loop is a variant of PID) for position, velocity and torque. This increases significantly their performance. To increase the performance of the controllers, offset values are added to the control loops (known as feed forward control). This way the controllers cope easier with their task (e.g. if acceleration is needed, the fed forward controller applies an offset to the control output of the torque loop the make the reaction time faster, the weight of the robot is compensates as an offset value on the torque coming from a static model, etc.). Industrial motor controllers have the same manufacturers as motors. Usually it is best to use a matching manufacturer.
Another difference is how the reference signals are given to the motor controllers. This is "only software" so it seems not expensive, but a lot of reaseach is in this domain. THe question is how to plan trajectories between two points so that the motor controller can deliver its best perfromance. You can read about this here.
Al of thea measure above is to "balance strenght and precision" as you formulated. Servos in hobbyist application are nowhere near their counterparts in industrial robots in terms of build quality and performance. I am afraid there is no cheap solution to get the same performance. The cheapest way to improve the performance of hobby equipment it though software. Get a servo where the cascaded control is implementable (like the dynamixel ones) and implement trajectory generators (many robotics courses are available online, like the link I sent you) that plan a trajectory while limiting jerk. Fine tune the control loop of the servo for your robot (instead of accepting the factory default). Compensate anything than can be calculated, eg. the static torque needed to hold the robot still as a feed forward controller.
With the above measures you can at leas get the most out of the hobbyist hardware, which will already make a difference.
