You have my sympathies Ploshkin - I've often bought fabric online only to find it's quite different from my expectations and unsuitable for the project I had in mind. I think "French Terry" is a broad church. I have 4 kinds in my stash, all completely different thicknesses and stretchiness. They all have looped backs though, two are cotton/lycra, the other two 100% cotton. I noticed that some online stores like Girl Charlee and Minerva are selling French Terry which is all polyester/viscose, so that must surely behave quite differently again. As Sewmuchmore says, I think loop back is the bottom line, although on one of my fabrics that's so small and fine you could miss it.
I rarely get swatches sent here 'cause I'm in Ireland but maybe that's your answer for future FT buys. By coincidence I read on a sewing blog yesterday about someone who made a great pair of Misty jeans from Fabric Godmother stretch denim then couldn't wear them because the fabric bled blue dye everywhere even after washing. She got a refund. If your fabric doesn't have a loop back, you could ask them in what way is it French Terry.
PS I'm about to make a Linden Sweatshirt from the thinnest of these. I can't see your fabric obviously but I've certainly made tee-shirts from thin, stretchy fabric and the neckbands and hems/waistbands worked fine (no interfacing). Yours may turn out better than you think!