Andrew, thanks for the elaboration - I had overlooked how the first law already refers to force. Probably the Galilean transformation follows in a similar way from the second law plus the first law. Note that since one century ago we know that the Galilean transformation is inexact, but that's beyond the scope of this forum.
And titaniumpen, as Simon remarked we certainly think that your original question has been answered*, we just continued the discussion as it triggered some more thoughts.
* we basically agree with you: your test allows to establish (for low speeds) that the trolley accelerates with a constant acceleration when you apply a constant force.
However, by using an old definition of force such as Newton likely used (for example kgf or kilogram-force), and with a modification of your experiment as discussed earlier, you can verify how well the second law of Newton works for accelerated trolleys: you should find that the acceleration is proportional to the impressed force and inversely proportional to the weight of the trolly.
In equation: a ~ F/m