there are people that say almost every MA (except the Simple MA) could be called "smoothed" - exponentials, Hull's, weighted - because there is involved some smoothing in the series. The exponential part of the EMA as we know it theoretically could be quite different too.
I am quite sure now that the "useless" version should be used for the Alligator. The Smoothed SMA differs in key moments from the other one (and the Alligator itself), so if it is used the indicator should not be called "Alligator". I guess people use the SmSMA to build the Alligator out of ignorance - most people who I have seen around "trying out the Alligator" look for a crossover of the averages - which has little value and by no means should be used for initializing a trade. Of course Bill Williams is the first to blame for he never wrote the actual formula in his books, but that is no excuse when money and dreams are involved.
I found and downloaded a version of "Investor's Dream" - the actual Bill Williams' software (it is full of trojans so I won't post a link here, however it was on some russian torrent and wasn't hard to be found). There was no option to add a moving average there on the charts or extract the formula, but the Alligator there looked the same as that one in Metatrader 4 - matched highs, lows, intersections with the price, fractals against it and graphic patterns for a couple of hundred bars on two instruments that I checked.
I compare to Metatrader because it is easiest for me to get historical data with it (Investor's Dream is quite old - the data included was from the 90's and I couldn't manage to import newer), not because I trust Metaquotes for their formulas at all (the Gann fan in Metatrader 4 for example is extremely incorect). But the Investor's Dream, the Metatrader and CQG all match the Alligators, and Metatrader SmMA matches CQG's SmMA and the Alligators so it should be correct
I am not sure if the formula for Exponential MA is the same across platforms - and I am almost sure that this is exactly where the next Alligator error will come from