Quantization error of SAR-ADCs
What the OP shows is the difference between truncation and rounding which in terms of error simply introduces an offset. They generate the same quantization noise spectrally; the standard deviation of the quantization noise (and equivalently noise power) is equivalents since DC offset is ignored in those computations. If there was concern with the offset error introduced (1/2 lsb) it can also easily be subtracted.
Further it should be clarified that DAC's and ADC's will have a finite DC offset regardless due to the analog portion of the electronics, this is often consistent with other DC offsets in the analog chain (consider the offset voltage between the negative and positive Op-amp terminal for example, or carrier leakage in zero-IF receivers), and when an actual issue (for a system where we are trying to measure a DC value) would be incorporated in an overall DC-offset removal algorithm with either calibration or cancellation loop) as part of all offsets combined. Since most DAC's and ADC's have a precision out to what there self-noise factors allow, the ENOB (equivalent number of bits) is often 1 to 2 bits worst than the stated precision (for example, a 12 bit converter may have an ENOB of 10.5 bits). For this reason it is often not even a concern if there is a DC offset introduced of 0.5 bits since that is well below the actual achieved precision overall. These details with regards to ENOB are clarified on ADC and DAC datasheets.
The OP asked for how the noise equivalence can be verified. It is obvious by inspection of the plots given that the only difference between the two is a static offset, but the equivalence can be computed simply enough for further verification by creating a floating point full scale  sine wave and then creating two quantization signals by both rounding and truncating the sine wave to the closest integer. Subtract the quantized fixed-point (integer) signal from the original floating point signal to get the quantization error alone. Then compute the standard deviation of each and they will be equivalent.
This chart I have demonstrates this graphically; specifically what is shown is generalized to a complex waveform (two ADC's with I and Q) but the concept is the same for a single ADC (real waveform), in that the rounded or truncated waveform is the sum of quantization noise with the higher precision waveform (such as floating point or more specifically samples of analog waveform prior quantization through rounding/truncation).

Further, if you use an incommensurate sine wave (choosing a frequency for the sine wave that isn't an integer sub-multiple of the sampling clock), you can also see through subsequent FFT and other statistical analysis how the quantization noise is well modeled as a uniform white noise distribution, and you can also confirm for either case (truncation or rounding) the formula used to estimate the SNR (the total power of the quantization noise floor relative to the full-scale sine wave) as given by:
$$SNR = 6.02 \text{dB/bit} + 1.76 \text{dB}$$
Which I detail further at this link:
What are advantages of having higher sampling rate of a signal?