You are correct.
And what I'm saying is that in head-to-head real-world tests, using tools like the one mentioned by Nate above, T-Mobile's speed was 3x faster than Sprint's "4g" network. Of course actual speeds weren't the "maximum possible" as listed in the marketing hype, but the ratio was still the same.
And of course, you'll need a new phone that can take advantage of the new technology, regardless of carrier.