If, while the battery terminal clamps are connected to the battery, you are reading different voltages between battery posts vs between battery terminal clamps, please skip down to WHILE CONNECTED below.
You are reading a very low voltage on the battery posts while the battery is disconnected. Then, when you connect the battery terminal clamps to the battery, the voltage goes even lower.
This is because your battery is nearly dead. When you connect the terminal clamps, you put a small load on the battery. This reduces the voltage even further. The small load could be from the dome light if a door is open or under-hood light; from the ECM, from the clock or the radio's clock, or from any number of electronic modules in the car that are always "on".
WHILE CONNECTED
There is only one reason why a battery post would give anything other than the exact same voltage reading as the terminal clamp while connected, and that is bad contact between the post and clamp. The bad contact can be due to oxidation or corrosion.
Also, under these circumstances, there will be a difference in voltage only if current is flowing. It could be a very small current in the milliamp range, but yes, current was flowing while you were testing. The ECM, the clock and any number of devices in the car that have electronic modules could be responsible for this current.
Repair: You need a wrench to loosen and re-tighten the clamps, you need a battery terminal cleaning tool and optionally, you need a small amount of greasy substance to exclude oxygen and battery fumes from the mating parts in the future. Vaseline works, but silicone grease or dielectric grease is better.
Best to wear safety glasses.
Remove only one terminal clamp at a time. Loosen the nut, grab the cable and move it back and forth to twist the clamp on the post until it loosens, then pop the clamp off. Don't let the battery clamp touch anything metal, or you could get a shower of sparks. (I usually put the clamp temporarily into a work glove.)
Clean the battery post with the cleaning tool. It looks like this:
Image: Harbor Freight. No endorsement intended.
The tool goes on the post while turning the tool. Rotate a few times, then twist it off. The post should now be clean and be shiny.
Now clean inside the clamp with the conical brush until it is clean and shiny.
Apply a light, barely-there coating of your greasy substance to the outside of the post and the inside of the clamp to prevent future corrosion. Install the clamp on the post (there may be a small spark), and tap it gently down with a block of wood or a plastic-tipped hammer until it is fully seated. Now tighten the clamp nut. Make sure the nut is good and tight. You should not be able to move the clamp by yanking on the cable.
Repeat with the other post and clamp.
Now when you measure voltage between the posts vs between the clamps, you will get the exact same reading. The reading may be slightly different if there is a heavy current drain or heavy charging in progress, but with the ignition off, it should be exactly the same on posts vs clamps.
