Is there a difference between measuring the resistance using a multimeter ohm function and using the voltage / current relation?
The Ohm function on your multimeter will use the voltage-current relation to determine the resistance of a sample. It will either apply a voltage and measure the current delivered, or force a current and measure the voltage required to deliver it. 

But it will almost certainly not use a voltage as high as 60 V, so it won't meet the needs of your test. 

Probably the ASTM spec requires a 60 V source to provide a repeatable test method that allows for breakdown of some of the thin layers of oxide material within the concrete, but not others. The higher voltage used may also be needed to accurately measure currents due to ionic transport within the never completely cured concrete.

Therefore I'd expect you need to use an actual 60 V source to achieve a compliant measurement.