Not exactly. Heat comes from electrons moving, which is Amps- so far so good. The way to get more Amps flowing is to either increase Voltage (motive force) or decrease Resistance. For example, an electric stove element has low Resistance, allows lots of Amps to flow from (typically) 220vAC.
An electric cigarette lighter in a car, has low resistance, and 12vDC input. Still gets hot.
Current (measured in Amperes) is generally represented in formulas by "I"
Voltage is "E" (sometimes V but traditionally E)
Power (measured in Watts) is "P"
Resistance is "R". There are additional impedances in addition to pure resistance, these are grouped together as "Z".
To get the power, remember "PIE": P=I*E
From this, you can derive the rest through basic algebraic substitution:
To get the current, I=P/E or E/R
Voltage E = I*R or P/I
The way motors work with voltage is a bit more complicated. You get AC motors and DC motors with different characteristics. For example, it's generally easier to make a DC motor variable speed, while AC motors are designed for particular speed settings or narrow ranges. With DC motors you can simply vary the voltage applied. Lower voltage, lower speed (obviously there is a maximum and a range where they are designed to work without excessive heat.)
DC motors have instantaneous torque with a pretty consistent torque across their speed range. Internal combustion engines have a much narrower power band where they develop maximum torque. This is why internal combustion engines stall out when load is applied at low rpms. AC motors have somewhat different torque characteristics than DC motors.
The main reason AC is more prevalent is because it is easier to transmit AC long distance at more consistent voltages due to the way transformers can regenerate the voltage (at a cost of drop in power gone to heat from resistance and impedance of capacitance and inductance qualities of wires and transformers) while DC voltage drops primarily due to the resistance of long wires, with loss in power also gone to heat. Some of the AC power loss (inductance in transformers) is used to "prop up" the voltage level for consistency of use by retail electrical devices designed for a narrow range of AC voltage. So transmission of AC electricity consumes some of the power but delivers a consistent voltage. This is kind of a "dumbed-down" explanation, and it has been 20 years since I formally studied electricity and electronic circuits in USAF technical school to get my Associate's Degree in Electronic Systems. I haven't tried to teach electronics, so I hope my explanation helps.
It's a worthy area of study for a backyard engineer.