If a dash was as much as 1/8 oz, then Johnson’s Martini recipe would contain a whopping 1/3 oz Boker’s bitters and 1/3 oz of gum syrup.
Regardless, 1/3 oz of lemon juice is less than half what winds up in the vast majority of citrus cocktails, today. My hunch is that we’re actually looking at a “lost style” of drink that went extinct once we could have all the fresh citrus we wanted and we got used to it.
I think a dash is an inherently subjective unit (dashers being inconsistent in design and inconsistent in yield, depending on fill level and vigorousness). “A dash” is seasoning, like salt in cooking (often not measured, but added in pinches based on experience). If these guys are writing down instructions in dashes, as opposed to measured quantities, then it seems to me to imply the use of dasher bottles and disretion. My interpretation is that Johnson treated lemon juice much like he treated bitters in many cocktail situations. Schmidt is another a-dash-of-this and a-dash-of-that mixologist.