I hear financial institutions declaring things like, “revenues have grown threefold over last year”. What they mean is this year’s revenues are three “times” as much as last year. Doesn’t “fold” imply a doubling? So that threefold would actually be eight times as much?
It would be my WAG that “-fold” is not a technical nor mathematical term. “Fold” itself may be, but not “-fold”. As such, they are using it in the vernacular and the vernacular definition wins (by default.)
I don’t know about mathematicians, but physicists often speak of “e-foldings”. This is in reference to exponential growth or decay, and refers to a power of e (the base of natural logarithms, 2.718281828…). As in, “The size of the Universe went through 60 or 70 e-foldings during Inflation.”.
This is consistent with the common usage you state: Going through one e-folding means an increase by a factor of e, and increasing 3-fold means an increase by a factor of 3. “Grown 3-fold” is not, however, synonymous with “Grown by three foldings”.
I believe it’s simply the same as ‘times.’ Because a two-fold increase is a doubling. A three-fold increase would be a tripling. I’ve never heard of a one-fold increase. 