Electronic – arduino – Tie Arduino – ATMEGA328P reset pin directly to +5V

IIRC the cap is wired to /RTS not DTR.

When /RTS is high there is 0V across the cap. When /RTS goes low the cap can not change voltage instantaneously so the reset line goes low. The cap charges through the 10K resistor. Once the reset line is above the reset threshold the uC program starts.

@LouisDavis referenced this page, which states

The ATmega168 is reset by pulsing its reset pin to GND. ...by setting the DTR line to LOW ... the reset pin gets sucked to LOW until the capacitor is charged through the internal pull up resistor and R1 - which resets the chip.

Simulated:

Auto-Reset
VDTR    1 0 PWL(0 5 0.5m 5 0.000500004 0 1m 0)
Vcc 3 0 5
C1  1 2 100n
R1  3 2 10k
.control
delete all
tran 10n 5m
plot v(1) v(2)
.endc
.END

V(1) is DTR or /RTS and V(2) is /RESET

The reset disable fuse does just that - disables the reset function of the associated pin. If you set that fuse, there is no way to reset the device except:

• Power Cycle
• Watchdog Timer Expiration

In-Circuit Serial Programming (also known as ICSP), I believe, relies on resetting the device by way of the Reset pin, and so you will not be able to reprogram the device once this fuse is set using ICSP. There is more than one way to reprogram an AVR though. If the device supports High Voltage Serial or Parallel programming (HVSP or HVPP) that's always an option. Or if you can include a bootloader on the chip (which can be done with an ATtiny85 for example, not sure about the ATtiny13), that can work too.

That being said, the datasheet section 24.3.3 has the following excerpts:

Special combinations of fuse bits will lock the device for further programming effectively turning it into an OTP device. The following combinations of settings/fuse bits will cause this effect:

• 128 kHz internal oscillator (CKSEL[1..0] = 11), shortest start-up time (SUT[1..0] = 00), Debugwire enabled (DWEN = 0) or Reset disabled RSTDISBL = 0.
• 9.6 MHz internal oscillator (CKSEL[1..0] = 10), shortest start-up time (SUT[1..0] = 00), Debugwire enabled (DWEN = 0) or Reset disabled RSTDISBL = 0.
• 4.8 MHz internal oscillator (CKSEL[1..0] = 01), shortest start-up time (SUT[1..0] = 00), Debugwire enabled (DWEN = 0) or Reset disabled RSTDISBL = 0.

If you have no intention of supporting in circuit programming, you can safely program once then set the reset disable fuse and use the pin's alternate function. You do not want to leave the reset function enabled in production unless you don't care about your device resetting in arbitrary ways. Basically if the pin is "low" for a given period of time, it will reset, end of story. Per the datasheet Table 18-4, the range of possible threshold values for external reset to trip is 0.2*Vcc to 0.9*Vcc... you maybe just be getting lucky with devices that are closer to 0.2*Vcc.