What is the correct way to interpret the built-in Analog-to-Digital Converter output?
the Atmega328P's datasheet, it indicates the conversion factor is
  5V/1024 [...]


Indeed. And the datasheet is the only authoritative source. All the
other sources are second-guessing.


  If Vin and Vref are 5V, then the ADC reading would need to be 1024 in
  order to have a true reading of 5V


You can't get a true reading of 5&nbsp;V. As per the datasheet (emphasis
mine): “0x000 represents analog ground, and 0x3FF represents the
selected reference voltage minus one LSB”.

Any input voltage larger that the maximum representable voltage will
read as 0x3ff (i.e.&nbsp;1023), just like any negative voltage (within
the forward voltage of the protection diode) will read as zero. Thus,
reading either&nbsp;0 or&nbsp;1023 should be interpreted as a saturation
of the ADC, meaning the actual voltage is unknown.


  The ADC is inherently sampling, and I am not sure if the ADC rounds
  up/down to the nearest value


In principle it should round to the nearest value. The transition in the
reading from&nbsp;0 to&nbsp;1 should ideally happen at
Vref/2048. C.f., again, the datasheet, subsection “ADC
Accuracy Definitions”: “Offset: The deviation of the first transition
(0x000 to 0x001) compared to the ideal transition (at 0.5 LSB).”

A little bit further in the page it is stated that the ideal position of
the last transition (1022 to&nbsp;1023) is “at 1.5 LSB below maximum”.


  Perhaps I am splitting hairs since the absolute accuracy of the ADC is
  "±2 LSB absolute accuracy"


Indeed. If you really care you should probably calibrate your ADC. See
Response of the Arduino ADC.

Edit: in a comment you mention that the factor&nbsp;1024 could
conceivably be “a persistent typo that it appears in multiple AVR
chipset datasheets”. It may be an error, but certainly not a typo, as
the datasheet contains multiple independent statements that confirm the
factor&nbsp;1024.