The only words we use for the [delineated] area designed to contain one car are "parking space" and "bay".
No, that's not true at all. We use 'bay' or 'space' for a single car.
I have heard 'bay' used in AmE as well. For me personally, a 'bay' would be a space in one of those covered parking shelters that don't have walls - just a roof to protect from the direct impact of the elements (mostly hail and to some extent the sun). These are fairly common in apartment complexes in the US, at least in my region. Often they don't have enough covered spaces for all residents, so renters pay a premium for the right to an assigned parking spot under the shelter. Each space is reserved only for one individual and violators risk having their vehicle towed at their expense.
I suspect at least some other AmE speakers might use 'bay' for a designated spot in an enclosed parking garage as well, though. My idea of 'bay' may just be my own.
Regardless, I tend to think of bays being reserved for one particular individual, either way - assigned parking, if you will. Again, that may be my own usage vs. general AmE usage.
The reason I put "delineated" in brackets in my previous post was to make it clear that, for me, a space doesn't have to have painted lines around it.
That's the same in AmE with both 'spot' and 'space'. It may or may not have painted lines delineating the space. In particular, parallel parking alongside sidewalks typically doesn't have marked spaces, although I have seen it. Regardless, an opening big enough to fit your vehicle would be a space (or spot).
It seems the notion of a 'lot' being a single space is limited to Malaysia and Singapore then, or perhaps SE Asia. Maybe I can try and remember to ask my Filipino friend next time I talk to her and see she uses it that way as well.