Question : Problem: Laptop not recognizing power supply, but power is going through.

This seems to be a very strange problem and it has been going on for sometime, I would not be surprised if the problem is the wiring in my home because I live in a college town where the homes are quite poorly built.

So, I brought my laptop from work back  to school with me and plugged it in to the wall via my trusted power supply. I worked on my laptop for about 5 hours and verified the power supply light was on in the front of the computer. II left it plugged in to my surge protector overnight, and when I woke up the battery had died and the power supply light was not on. I unplugged everything and took the battery out, put it all back together and still no power. I asked someone at work to send me a new PS, and when I received that, my laptop and power supply light were up and running again. I left it plugged in overnight --this time I made sure it was plugged into my switched power conditioner-- but once again, same problem.

I hooked up my voltmeter to both power supplies. Both are reading 120VAC in from the wall and 18VDC out of the PS.

Another seemingly related issue is that my other laptop's power supply, a Mac Powerbook 65W adapter, loses power every once in a while. I notice that the light on the power supply goes off and I have to unplug it from the wall, not the computer, to get it to turn back on.

Answer : Problem: Laptop not recognizing power supply, but power is going through.

Sounds like a very strange problem if it is happening to two DIFFERENT kinds of laptop power cords...IE: HP/Compaq  verses Mac Powerbook.
You might check to see if the JAck (dc jack) where you plug in the power cord to is not lose or broken.  that is one of the most common problems with laptops is the dc jack gets lose or broken.  
What usually causes this is if the laptop cord is pushed into the jack and the cord gets tweaked or the laptop gets dropped or pushed up against a wall and the cord pushes on the jack.  Sometimes the solder joints get broken easily with heat and the torque to the cord.  I suggest taking a small fine tipped pen or screwdriver and see if you can wiggle the DC Jack in the back of the laptop.
See if the jack is lose?  etc?  If you have to move the cord a certain direction to get the laptop to power up or charge, then that is most likely the problem.  Let us know what you find out.
IF you still can't figure it out, sometimes you can take the laptop to your local repair shop or sometimes even batteries plus and they can test the battery.  Which could also be that the battery is shot and wont hold a charge...
Another test is to see if the laptop will power up with out the battery installed, and running off the power cord by itself...