both my laptops have ATI Radeon onboard cards
Why oh why did you do that?? 
Oh well, still better than an Intel GMA.
(the Toshiba runs better probably because I haven't run it into the ground like the HP, and also because Alien Swarm, Portal, and Sins of a Solar Empire are much less stressful than, say, Dawn of War 2)
Yes, there is a lot of difference between each video game you make your computer run. 1 game might fry your GPU, with one other might keep it cool & quiet. For this reason, and because you always want to test the reliability of each new hardware, when you get a new laptop (or desktop, whichever):
1. Install drivers, DirectX, and all necessary software.
2. Install Furmark's latest version.
3. Run Furmark at "Stability Test" mode, for 20-30 minutes.
4. Observe and note the highest temperature reached.
5. Close Furmark and compare your temps with other similar hardware online. (preferably report those temps here. I can tell you if they are normal)
Overall, a GPU, whether it is an on-board chip, or an add-on GFX card, SHOULD run Furmark without any issues, at stock clock speeds. If not, then either the hardware is faulty, and/or bad, or your computer runs hot inside, thus affecting your GPU's temp.
Dual core 2.4 GHz processor, 1GB Nvidia graphics card w/ 1274MB dynamic memory allocation (total 2298MB gfx mem), 500GB hard drive, 4GB RAM (expandable to 8GB)
Thats doesnt mean much, does it? Your CPU could be a Core2Duo at 2.4, or an AMD Athlon X2 at 2.4, or.....you get the point.
Same goes for the GFX card. There are many Nvidia cards that have 1GB RAM you know. Btw, frame buffer (or graphics RAM) plays a very small part performance-wise, regarding GFX cards. And unless you play at resolutions higher than 1920x1080, then you should not care about graphics RAM at all. It doesnt have a significant positive or negative impact on performance.
Also, the dynamic memory allocation is nothing but marketing crap, designed to make people think, that even with a cheap GPU, they can run games smoothly and allowing their GPU to run faster than it normally could. Sadly, this isnt the case. When you have a "weak" GPU installed, increasing frame buffer wont help at all. Today's Graphics cards arent limited by frame buffer. That was the case 5+ years ago, when we had Ati Radeon 9800s, and 256 MB of Ram still wasnt enough. Back then, every little MB of ram helped a lot.
Today, its nothing more but a marketing scheme. If you want to have 3 screens, with 2560x1600 resolutions or something, then sure. 1 GB isnt enough for you. But if you had so much money to spend on 3 freakin huge screens, why on earth would you buy a cheap GFX card anyway?? So normal people typically dont need more than 1GB of graphics dedicated Ram.
I forgot to mention that the dynamic memory allocation on these GFX cards, is basically your card "borrowing" some of your system Ram. In your case, your GFX card will borrow 1.3 GB of your 4GB of system Ram, to keep frame rate up, in case you wanna run a game at extreme resolutions or something. (Which btw, helps performance very little, since system Ram isnt as nearly fast as GFX Ram)
Othersise the specs seem fine.
And an 18.4" screen. The thing is though, I don't need it to play any of my gaims at high detail/resolution. I'd gladly settle for Dawn of War 2 Chaos Rising on all low graphics settings (which is what I had to use anyway), so long as it ran as smooth as an oiled, ah, well, female form?
Anyways, since DoW2 is the most graphically intensive game I own, if that runs on low (or medium) settings, than all my other games will do perfectly well I'd imagine.
Oh god, Low graphics settings..... 
Ah well, I guess there are some people that dont care much about that. Me? I wouldnt settle playing a game at less than High gfx settings for anything.
I dont mind games with poor graphics even if they are maxed out, but I do mind having to reduce graphics detail on games with good graphics. Makes sense?
RE: Gaming laptops: It is really funny what stores/shops call "Gaming laptops" today. They sell a laptop with a GeForce 430, and they call it "Gaming", LOLZ 
So hilariously stupid.
Ah anyway, here is a REAL Gaming Laptop:
http://www.newegg.com/Product/Product.aspx?Item=N82E16834214053
But this is what I would buy IF I wanted a laptop and use it for gaming:
http://www.plaisio.gr/Laptop-Netbook-GPS/Notebook/Laptop/Turbo-X-Dominus-480.htm (Its in Greek, but you can read the specs, and watch the graphs.
Overall, laptops are not made for gaming. Speaking of laptop GPUs: Even some popular models by ATI and NVIDIA are not officially supported by most modern games. That is basically the companies (ati, nvidia) saying: You can use that GPU, but we cant guarantee that it will run games smoothly and without any issues.
Last but definitely not least: Get a desktop for gaming only. Do everything else besides gaming on your laptop. Play games on your desktop PC. Have fun.
"We were inhuman beings....butchers on a field of corpses....corrupt and depraved....only a sum of viscera, blood, and bone - Creatures without soul."
