something like this installed where the driveway meets the garage slab:

Here is a link that might be useful: drainage grate

sierraeast, what do you mean by "something like this installed where the driveway meets the garage slab"? Can you explain it?

Ok, I saw the link about Trench Drain. Do I have to dig out driveway to place a Trench Drain, by the way?

Yes, you actually need to saw cut the concrete, (rent a concrete saw from a rental yard), the width of the grate per installation instructions. You will also need to have a drainage system at some point where the water in the grate is directed to,( such as a french drain system). The referall was the only website I saw when googling, but there are many grate systems out there. You might try a reputable masonry supplier in your area and see what they have/recommend.

I looked again at your outside picture. You could divert the drainage grate, (or trench grate as I believe they are called), over to where your downspout is draing down and out, tying it in to that. You would run a catch basin at that end of the trench drain under the downspout with a grate cover there and divert it out and away from the foundation/driveway by way of a french drain type system.

Looking at your inside picture, do I see sunlight under the door? Could it be as simple as continuing the rubber gasket all the way to the edge?

The gasket is not long enough.

It should go all the way to the very edge of the door.

I think you should use some metal sheet for that, i hope that can work better to prevent your water.

Here is a link that might be useful: Liquid Roof

Buy a new gasket and replace the whole thing.
Ron

I never did that before. Is it easy to replace the rubber gasket under the car garage door, by the way? Are there any tips to buy and install the gasket?

we had the same problem except worse because our garage is at the bottom of our steep driveway so the rain was rushing down the driveway and under the garage door making a huge pool in there. We did have a drainage channel in front of the garage but the water couldnt drain away quickly enough so it just overflowed, and all we did was install a french drain at the side of the driveway where the water could drain off to and it completely fixed the problem. By the way, our drainage channel isnt one you buy, it's just a concrete half tunnel that is part of the driveway so you could easily make one after cutting a small strip out

the rubber gasket at the bottom of the door is primary to stop drafts, and to dflect water away. It is not a waterstop for ponding water at the base of the door. Proper technique to drain away from the house. As far as difficulty replacing the rubber it depends on the door type and how the original one is attached.

Daylight under the door might also be because the top of the slab is not parallel with the door or maybe the door is not tracking perpendicular. I'd bet if you just pushed down on the door slightly after it closes the gap would be closed too.

HTH

"the rubber gasket at the bottom of the door is primary to stop drafts, and to dflect water away."

It keeps out casual rainfall very well.

While it will not work for ponding water, the leak appears only were the gasket is not present (and light is showing).