The recovery partition on your original disk was the only supported method until the Recovery Disk Assistant was made available.
The bulk of this answer was written before the official steps were published - so I would recommend the Recovery Disk Assistant even though HT4718 steps help understand what is happening under the hood.
http://support.apple.com/kb/HT4718
The easiest is to get a drive enclosure or adapter and re-run the Lion installer from the Recovery HD and install Lion onto the new SSD.
It likely won't matter if the drive with the recovery HD is on an external bus or an internal bus, but I would keep the original drive inside the mac until you have Lion installed on your SSD connected to an external port.
Boot off the SSD to test it and run migration from the internal drive to the SSD before doing swapping the drives inside your mac.
It still boggles my mind that any mac will ship without some sort of reinstall media. Look in the center of the box, under where the mac sat. Call apple if your media is missing so they can ship it to you or point out how they want you to reinstall the OS on a new drive. Your mac's hard drive is listed as a consumer replaceable part, so they have to provide you with the steps to do that.
There is definitely something wrong, what you've described sounds like exactly the right steps to solve your problem, either you're doing something wrong or the disk is damaged or your Mac's optical drive isn't working.
Hold down the option key while booting your Mac, and insert the Snow Leopard install disk. It should show up after a minute (maybe you have to press "refresh"? I can't remember), and you can boot from it. Before installing, it's probably a good idea to use Disk Utility to repartition and erase your hard drive, specifically to remove the recovery partition that Lion has created for you (it will be recreated if you install Lion again later).
Failing that, you can try booting from the Snow Leopard install disk using another Mac to see if it's working, or try using someone else's install disk on your Mac. You might also try duplicating the disk on your Windows pc (using an app that does a low level/raw disk copy, not just a file copy — which will not work).
If none of that works for you, perhaps take your Snow Leopard install disk in to an Apple store or Reseller and tell them you can't get it to boot, maybe they will be willing to test the disk for you and perhaps give you a new one (for free if they're really nice).
You can also use someone else's Mac to download Lion (open the App Store on their Mac, log out of their account, log into your account, go to Purchases and click download on Lion, then log out of your account. Remember to log out of your account! otherwise they might accidentally buy apps with your credit card). Once you have the Lion install package on their Mac, right-click and Show Package Contents on it and drill through the folders until you find the InstallESD.dmg file. Use Disk Utility to copy that DMG onto a DVD or USB memory stick (use the "backup/restore" feature to copy the DMG's content onto the disk). Old Macs can't boot off a USB memory stick, but all recent ones can.
I've been told most Apple Stores are willing to do the above for you if you explain you've got a slow internet connection at home.
Best Answer
Oh goody a classic MacOS question...
The nice thing about the classic MacOS is that once it is installed you need little else besides the files in the system folder. Yes there is the "blessing" of the system folder but if it is your intention to use it in a "Classic Environment" inside macOS then you should not need to worry about that.
Connect the drive with the macOS Tiger installation and boot from that. I think that will work. Some older Macs cant boot from USB. Failing that can you put it in an old firewire enclosure?
The goal here is that once booted into macOS Tiger just drag the folders from the classic install on the SSD to your Tiger drive. Put them at the root of your (Tiger) Mac HD and take the whole system folder, applications folder and anything else you need.
I did this a while ago and while I was at it I made a zip archive of the whole thing, "just in case." Just in case happened and I unzipped the files and it worked a treat.
Once you have done that you can use an old (shareware) copy of Carbon Copy Cloner to "back up" the Tiger drive with MacOS 9 on it to the internal SSD.