Short Version: How exactly do (modern) voltage/current meters measure the numbers they do, from first principles? References much appreciated.
We measure length with a ruler, time with a clock, and mass with a scale. The units of such quantities are all relative to some commonly accepted standard, i.e. a stick of a fixed length, a certain interval of time, a certain amount of water at fixed conditions, etc. These quantities are relatively easy for us humans to understand.
But what about voltage and current? I understand how you can define electric fields experimentally - get a certain amount of charge on an object, and measure the force (response) felt by that charged object. Similarly with magnetic fields - send some charges through a magnetic field and measure their response. To measure the voltage across two points, I suppose you could connect some small "resistor" in between two points and measure how hot it gets under certain fixed conditions (maintained within your measurement device).
I'm sure my intuition on all this "phenomenology" of voltage and current is based on 19th century equipment, but that's the only equipment I ever see intuitively broken down. Modern electrical equipment is more mysterious (owing to its inherent complexity I'm sure).
Every time I try to look for an explanation of how multimeters really measure voltage and current, I'm met with circular explanations like "You measure voltage by connecting a resistor in parallel and measuring the voltage drop across it, or current passing through it" and "You measure current by connecting a resistor in series and measuring the voltage across it". Please, I just want an intuitive and direct answer. I would also very much appreciate a reference to some article/discussion on this.