I agree with the precision (I thought it was +/- 3%).
The higher moisture content reading when you push the pins in is because of the following.
The meter does not measure moisture. It measures electrical resistance. Moisture decreases electrical resistance in wood. So the lower the resistance, the higher the moisture content reading. They have a calibration table loaded in the device to transform the resistance to a moisture content reading.
Now, if you push the pins in deeper, they will sample more wood. I.e. more parallel current paths for the electrical current to flow. I.e. a lower total resistance. And hence a higher moisture content reading.
Second, the deeper you go the more wet the wood is; if you only touch the (dried out) outer surface, you'll also get a lower moisture reading.
So, it's always best to push them in as far as you can. Without bending them (when you pull them out). And some (oak) times that's just not possible. So be it. You'll eventually figure out how the splits will burn when a reading is X and how they will burn when the reading is Y. And then you'll know how long it takes to get there, how the splits feel and sound, and you won't need the moisture meter anymore.
For now, a reading below 20% is best. In fact, it's nice to collect some <20% wood and some e.g. 23-24% wood. Have a bed of coals (assuming this is a N/S loading stove), rake them forward, load the left half of the firebox with the drier wood and the right half with the wetter wood. You'll see a big difference.
(Don't do this too often as the wetter side will produce creosote - but for a one-time educational experience, it's okay.)