I would agree with the basic idea that there are three levels of pain, and also that only great apes are aware that they are in pain.
In fact humans may be in pain, but not be aware of it. I recently had a moderately serious accident, and cut my thumb deeply ( the tip of the bone was sliced off, to give you an idea ). I then probably cycled home ( I don't remember that well due to concussion, of which I was completely unaware ), and was quite unaware that I was in pain. I did know that I had cut my thumb. You might argue that I wasn't even in pain, that's debatable.
I would also cite the example of young babies - they have very little self-awareness (I don't recall the age at which it develops, but it is I think after birth), but can you assert they do not suffer when experiencing pain?
Regardless, the big jump here is going from "animals (other than great apes) not being aware that they are in pain" to the title of your post which is "An argument that animals don't really suffer". Why is suffering related to awareness of being in pain? Isn't it enough just to be in pain?
It's quite likely that you weren't in level 2 pain at all - endorphins will do that to you. It would be very different if you had started feeling pain and then decided to use concious meditation in order to mitigate it. Roughly stated, you would feel your assessment of the sensation change from "pain/suffering" to "meh... some weird stimulation I don't really care about".
I ended up reading this article about animal suffering by this Christian apologist called William Craig. Forgive the source, please.
He continues the argument here.
How decent do you think this argument is? I don't know where to look to evaluate the core claim, as I know very little neuroscience myself. I'm quite concerned about animal suffering, and choose to be vegetarian largely on the basis of that concern. How much should my decision on that be affected by this argument?
EDIT: David_Gerard wins by doing the basic Google search that I neglected. It seems that the argument is flawed. Particularly, animals apart from primates have pre-frontal cortexes.