In NEHRP Provisions there is an interesting commentary that gives a reconciliation between 0.2 and 2/3 coefficients.

The title is NEHRP Recommended Seismic Provisions for New Buildings and Other Structures. The last edition is commonly referred as 2015 NEHRP Provisions/ or FEMA P-1050-1 and can be found on internet as fema_p-1050-1.pdf

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Although no specific rationale or logic is provided in editions of the NEHRP provisions (FEMA 2009a) on how the value of 0.2SDS was determined, one possible way to rationalize the selection of the 0.2SDS value is to recognize that it is equivalent to (2/3)(0.3)SDS, where the 2/3 factor represents the often-assumed ratio between the vertical and horizontal components of motion, and the 0.3 factor represents the 30% in the 100% to 30% orthogonal load combination rule used for horizontal motions.

Although details regarding defining vertical ground motion spectra are currently well known, the committee elected not to define a vertical ground motion spectra in this standard because the approach provided by the equivalent static coefficient 0.2SDS is adequate. For situations where one wishes to explicitly include the vertical component of ground motion in design analysis, one may use the vertical ground motion spectra definition that is provided in the “New Chapter 23, Vertical Ground Motions for Seismic Design” in the 2009 edition of the NEHRP provisions (FEMA 2009a).
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BR