The Saturn V (Apollo moon rocket launches) were somewhat different from the Space Shuttle, with very gradual initial acceleration (seen in contemporary footage). Most of the thrust is used to overcome the sheer weight of the rocket, and only once the propellant begins to burn off, does this lead to increasing acceleration (given a constant thrust). This process might be part of the 'myth' that you quote.
The first stage of the Saturn V generated around 7.6 million lbs of thrust at lift-off, with a total stack weight of around 6.8 million lbs (figures varied slightly from Apollo 9 to 17). The STS SRBs generated 5.6 million lbs thrust and the three SSME's generated another 1.2 million lbs, pushing a total stack mass of around 4.5 million lbs. So it's not surprising that the STS was accelerating much more smartly at lift-off; its thrust-to-weight ratio was quite a bit better at launch.
On the other hand, toward the end of the Saturn V first-stage burn, the center engine would be cut in order to limit acceleration to under 4Gs. By this point the vehicle had burnt off close to 4.5 million lbs; ie it was less that half the weight it had been at lift-off 2.5 minutes previously.