Not sure if this has been shared here yet, so I will share it with you all. I stumbled across a great guide elsewhere on the net on how to play PS1 backups on a PS2 - even those of a different region to your own. This has been written for and tested on a slim (SCPH-77002) PS2 - I'm not sure if it works on other models.
Just a note in advance: this method of playing PS1 backups DOES NOT require you to swap discs while the motor is spinning. Every time you swap a disc, the motor will be stopped. Therefore, it's a relatively safe method of playing PS1 backups.
First of all, a couple of things you will need - you will need FMCB and uLaunchELF loaded on to your memory card. If you are unsure of what this is or how to do this, look around on these forums - there are plenty of tutorials and free services available to help you.
You will also need to block the PS2's lid sensors. There are countless directions & video tutorials on the net on how to do this - the specific one I used can be found here: link removed. please don't use tinyurl ~bootlegninja
Once you have FMCB and uLaunchELF loaded onto your memory card and you have blocked your lid sensors, load up the FMCB configuration tool and assign uLaunchELF to an on-boot button - I used R1.
Next, fire up uLaunchELF, enter the configuration menu, and ensure "Disc Control" is set to ON.
Now, for PS1 backups of the same region as your console, place any original PS1 game in the disc tray, close the lid, then turn the console on while holding down R1 (or whichever button you assigned uLaunchELF on-boot to).
Once uLaunchELF loads up, your disc will stop spinning. Open the tray, remove the disc, replace it with your backup, close the lid, then select "PS2Browser" from the uLaunchELF menu.
Wait a few moments and voilą - you're playing your PS1 backup on your PS2!
For PS1 backups of a different region than your console, you will need an original or a backup of the Action Replay 2 v2 Bonus Code Archive CD (I've been informed other code CD's can do the same thing but have not tested them myself). You will need to make sure that the copy of the CD that you have is of the same region as your console (PAL, NTSC etc.)
Place any original PS1 game in the disc tray, close the lid, then turn the console on while holding down R1 (or whichever buttot you assigned uLaunchELF on-boot to).
Once uLaunchELF loads up, your disc will stop spinning. Open the tray, remove the disc, replace it with your Action Replay 2 v2 Bonus Code Archive CD, close the lid, then select "PS2Browser" from the uLaunchELF menu.
Wait a few moments and you will be greeted by the Action Replay 2 v2 menu. Navigate through the menus until you find "Start Game Without Codes" and select it. It will instruct you to swap discs, so go right ahead - open the tray and remove the Action Replay 2 v2 CD (again, it will have stopped spinning so you don't risk damaging your console by hastily swapping) and replace it with your different region PS1 backup.
Press X, wait a few moments, and voilą - you're playing your different region PS1 backup on your PS2!
I hope that this helps people who wish to play PS1 backups on their PS2 without getting a modchip or similar. Feedback is welcome, but please don't abuse me.
Thanks to Polosolo for the original guide.