What can be done wrong changing the chain in a single speed? I changed the chain of my singlespeed. I though it was going to be something really simple and quick, but I was completely wrong...

I keep getting a lot of noise while I'm pedaling (unless I go really slow/downhill) 

What I've done:


Remove the old chain (It had around 1500-2000km).
Measure the number of links and cut the new change to the same number.
Close the chain.
Adjust tension so it can be moved around 3-4cm in vertical direction.
Center the wheel.


The same in photos:

I ordered a singlespeed chain and I think it is the same as what I had, at least I don't see any difference. The silver chain is the new one:





Installed:





Tension:





If I added tension the pedals would not move freely backwards, if I remove tension it feels too loose.

I tried first without lubing, and then when I heard so much noise I added some oil but didn't make much of a difference.

What am I missing? The noise is the all the time when I pedal in in the whole range of motion.

Added photos of the cog and chainring:
The photos show that the contact surfaces of the cogs have been worn to match the old chain. The new chain does not match them perfectly, which is the reason to the noise. 

For some reason singlespeed seems to be much more sensitive than derailleur cogs. The good news is that if you can tolerate the noise and vibration for a while, the new chain and old cogs do wear to match each other. If you do not want to do this, replacing the freewheel should help.