Do you still have/know the source of where Sony calls the PS2 Slim? I was having trouble locating it at the first glance in the Internet, and nor the box or the manual I have did not said anything about it at the first glance either (it's from the latest version), though I don't have a box of the earlier Slim PS2s to confirm (but a glance at box images on Google didn't said so either).
Sony uses the terms 'slim' and 'slimline' for many of their products. It is a description of a design or a redesign. No where can I find it being used as a product name. Other things they use slim with are VAIO notebooks and televisions.
After some searching, I think tripredacus is right. I knew I saw Sony mention them, but I didn't pay attention to the actual wording on systems, my bad
I guess we'll have to think this over too.