Uploading a skectch into atmega328 that has uno bootloader and is in circuit

I'm thinking what you guys are describing is, e.g., the breadboard circuit in this tutorial (with arduino pin 10 to reset, and pins 11, 12, and 13 the serial programming pins) am I right?:

If so... I'm a little confused on what exactly is happening differently (in context of Arduino-as-ISP) between writing programs to the chip on the Arduino board vs. writing programs to the external chip?
I'm guessing it's related to Reset... what precisely is the function of reset here?

Based on what I've understood, reset serves to trigger the microcontroller to re-initialize, thus be open to (with the original bootloader still in place) new uploads, or if nothing is incoming, just lets it re-run the existent program in its memory. So, when @retrolefty suggested grounding the Reset pin on the arduino while loading programs to an external chip, what does that do?
(Reset is active low based on datasheet, so I understand somewhat...)