Q: The definitive method to probe the device operating current How can I measure the current a device consumes?
I tried one that has 1.5 V power supply requirement.  I measured its resistance with a DMM from the power supply box (with the batteries removed) by connecting the two poles.  It read 1.41 Mohm, so I thought the answer is more than 1 microampere.
When I put the DMM in series with the power supply (with batteries installed) reads  2.45 mA i.e. gives device resistancr about 0.6 kohm (1.5V / 2.45 mA.)
What is the correct method to measure the current?

A: A current meter in series between the device and the power supply is the correct way.
You have to provide the proper voltage and current to the device for it to power on and operate normally.
The resistance mode of a DMM can't supply the correct current or voltage to the device so it doesn't actually turn on and operate.
The current you measured with the meter in current (ampere) mode is correct.

Modern devices might not be simple to measure.  They turn on and off, and "take naps" to reduce the total power consumed.
For that kind of thing you may need special equipment to measure fast changes in the current.
For a simple "how long will my LED torch run on batteries," a measurement with a DMM is adequate.

A: If the devices is in a well defined steady state and the current consumption doesn't vary by decade, than you can use Agilent 66319D


If your device is consuming very little power and stays in that state for long, see this post:
How to Measure Power Consumption on Extremely Low Power Devices?

Measuring dynamic currents from nA to A is difficult.
If your devices goes, many times per second, from sleep mode to operating mode and the current changes by more than a decade than you need an expansive instrument like this:
Keysight N6781A
