Ok, so I'm slightly confused on what your biggest priority is. Beater and ride not to worry about driving aren't really synonyms.
If you just want a reliable ride to get around town in, then you'll be fine with an Accord or a Civic. It's my personal believe that Accords hold up to the tests of time a little better than the Civs, but that's an opion. I also much prefer the Accords.
However if you want to have a little fun and trick it out, there's tons of stuff for Civics. It's a lot easier to, say, get a turbo for a Civic than an Accord. Especially the newer Accords, but you didn't include those in your year lists. I'd recommend a Civ over an Accord if you want to turn it into a toy and squeeze some juice out of the powerplant.
It really just depends on what you want, though. The Accord is definitely a more "classy" ride, has some better stuff in it (such as sound-deadening material), and holds more junk if that's your bag.
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