I tried to take my old Italian u-folder on the SS last Summer, and the conductor kicked me off because it wasn't in a bag. It doesn't really fold small enough for a bag.
So I went back to the desk to get a refund, and the saleswoman said he shouldn't have done that because I was inside the timeframe for bikes to be allowed.
So firstly, it's down to who you end up dealing with, but secondly, I would get a smaller modern folder that fits in a bag. I'm not that well-versed on which ones to get, but I bet some of the other folks on here could suggest some good models for you.
Brompton: small, compact, great for trains, fits in a bag or pseudobag with no bottom easily, but a bit expensive, and gearing is odd, though usable. This is ideal for your intended use, and you can carry a bunch on it with the front cargo bags they have. I'd get one with fenders, no rack, 3 or more gears.
Bike Friday New World Tourist and similar: disassemble-able, don't fold fast, clunky when folded, but will disassemble into a suitcase that later works as a trailer. Good for planes, not trains.
Bike Friday Tikit - a bit clunky but possibly usable, needs to be held closed.
Birdy: again, clunky when folded. I don't think the ones I call clunky will fit well on a rack since they have sharp bits sticking off at odd angles and they don't hold themselves folded as you lift them up and down.
Dahon: many versions, most have a single fold and are clunky for putting on an overhead rack. Some might be OK - I don't have one. There is supposedly a newish Dahon that uses the Brompton type fold, nearly, perhaps not easy to find. There is a new Dahon spinoff company called Tern mentioned elsewhere in the folders group. I think these will all need a bag to work on a train, which costs you time. Inexpensive and well regarded bikes though.
Xootr or other Swift: big package when folded, don't think it would convince many conductors.
Strida - would fit in a rack easily - it's like a big broom - but not supposed to be nice to ride (haven't tried it)
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I've taken my Dahon Speed (20" wheels) on the South Shore a few times without incident. It's been bagged, and I put it on the seat next to me. I had it ready to go before the train arrived, and the conductor did not give me a problem on any trips.