The 8051 microcontroller is a microprocessor with RAM, ROM, I/O ports, timers, and serial ports on chip.
Is that the correct distinction between a microprocessor and microcontroller?
It is very common to mix up microprocessors with microcontrollers. Most 8051 variants I've come across would be classified as microcontrollers.
Typically a microprocessor is defined as a CPU core and not much else. This would consist of circuitry comprising an Arithmetic Logic Unit (ALU), Program Counter (PC), Stack Pointer (SP), and various registers to hold the data being operated on. Instructions and data are loaded into the registers via buses from external memories. The CPU in your desktop or laptop would be considered a microprocessor.
A microcontroller consists of a microprocessor and various associated peripherals. These may include (but are not limited to) Memory (ROM and/or RAM), General Purpose IO (GPIO), communications interfaces, timers, PWM generators, and capture/compare circuitry. AVR (Arduino), PIC, MSP, and most 8051's are considered microcontrollers.
Things can get even more confusing when you bring Digital Signal Processors (DSP) and System On a Chip (SoC) into the picture.
DSPs are a special form of microcontroller that typically have special instructions integrated into the microprocessor core to handle floating point numbers or specialized multiply instructions to make the programming of certain types of digital filters more efficient. They also usually have Analog to Digital Converters (ADC) or Digital to Analog Converters (DAC) listed among their peripherals. TI's C6000, Freescale MSC, and Analog Devices SHARC are examples of DSPs.
SoCs consist of one or more microcontroller and/or DSP cores, a variety of on-chip memories, programming interfaces, and timing sources. Oftentimes they include Radio Frequency hardware on the chip such as can be found in your smartphone. It is also common to see voltage regulation and circuitry to monitor power and temperature on the chip as well. You will occasionally see programmable logic incorporated into the design as well such as in the Cypress PSoC.