...I didn't see your image either, but I was able to "find it".
One of your "problems" is, 3V is not enough. You are operating the LEDs at "undefined" conditions.
LEDs are a little tricky because (like all diodes) they are non-linear. That means the resistance changes when voltage changes. The resistance is high below the forward operating voltage.* Above the operating voltage, resistance is low.
Red LEDs operate at about 2V, so you need at least 6V to "properly" run 3 of them in series.
Voltages across series components sum-up. Or we can say the voltage "divides" among the series components, proportional to their resistance. If you apply 3V, then the individual voltages will sum to 3V. (If you measure something different, you're getting measurement errors with your meter, and no meter is perfect.)
So normally, here's how an LED & resistor work - Because of the non-linearity of the LED, it operates at "constant voltage". If you increase the voltage, current increases and the LED gets brighter, but since the resistance of the LED drops, you only get a very-small voltage increase across the LED and most of that increased voltage ends-up across the resistor. In other words, the correct voltage across the resistor "falls into place". (Except in your case you don't have enough voltage for 3 LEDs.)
When we choose a resistor for an LED, we look at the LED specs and available voltage. We subtract the LED voltage from the power supply voltage to determine the voltage across the resistor. Then, knowing the voltage across the resistor, we calculate a resistor value to give us the desired current.
and current measurements
If you are using the current-range on your meter - It's better to measure the voltage across the (known) resistor and use Ohm's Law to calculate the current. (The current is the same through all series components.)
Measuring current with a meter is tricky because you have to break the circuit and insert the meter, and it's "dangerous" because in the current-mode your meter is a short circuit... If you accidently connect it wrong you can blow the fuse in the meter or fry your circuit under test. I measure voltage & resistance every day at work, but I almost never measure current (except that my bench power supply has an ammeter and I do keep an eye on that).
- On a "regular" diode, this is the forward "breakdown" voltage, and it's about 0.7V for a regular silicon diode.