none of the above.

Take a look at Sandhill Preservation's website for Cucumbers at:
Sandhill Cucumbers

The Japanese Climbing is a general purpose cucumber that produces an amazing number of fruits.

Tokiwa is probably closest to what you are looking for.

Look through the rest, there are some really unusual cucumbers in that list.

DarJones

It depends on what you want for flavor, how many seeds you will accept, and how important it is to be bitter-free.

"Straight Eight" is a great cuke; but it has a fairly large seed cavity, and IMO, is best picked young. It was mildly bitter for me on occasion, but this is minimized if the plants are kept watered.

Many of the Asian varieties are burpless, and generally very low in bitterness. Long & thin, too, much like the English cukes. Suyo (there are several spellings in circulation) is a good commercial cultivar. Yamato Extra Long is another.

I've witnessed SSE's Nippon Sanjaku Kiuri on their farm & it was highly productive with long, skinny cukes; but I have haven't tasted it.

If you like crunchy & bitter-free, try Armenian. Outstanding flavor when picked young. It is actually an elongated melon, though, bred to be eaten immature. The "cukes" are hairy, not spiny.

My personal favorite is not exactly an heirloom, although I doubt it can be found commercially. WI 5207 (developed in my home state) is a strange bird... an open-pollinated parthenocarpic slicer. If grown under cover it will produce large numbers of seedless cukes (they have tiny, undeveloped seeds). Even if pollinated, there will be few seeds, and they will have a small seed cavity. The flavor is very mild & low in acid. The vines are also resistant to a host of diseases.

They don't have to be long and thin, but I do want them to taste like a regular cucumber, nothing too radical. Do the asian cucumbers taste like what I would imagine a cucumber to taste like?

We do tend to go away for several days at a time in the summer, so I need something that won't get bitter right away if it doesn't get watered every day during hot weather. Bitter-free is fairly important. I water fairly regularly, but it seems that some varieties get bitter faster than others and I need a bit of leeway :)

Seed quantity isn't as big a deal... I just dislike the taste of biting into a cucumber slice and having it taste "seedy", you know? As long as it isn't overwhelming, that's not as big of an issue.

I'm not sure about the Melon one ... I've actually read about Armenian online but I'm worried it wouldn't taste "cucumber"-y enough for us. Kids, especially, are less likely to eat a cucumber if it doesn't taste enough like a cucumber!

Also I should mention that another important criteria is a thin enough skin to eat ... without having to peel first ... if possible.