There's a more detailed model of the experiment in which you aren't making the same measurement twice. You say it yourself, the two measurements are made at different times. It's better, IMO, to get out of thinking in terms of the first approximation that is quantum mechanics, instead moving to thinking in terms of the second approximation that is quantum field theory.
In a quantum field theory, there is a difficult question of just what you might be measuring. If we think in terms of QFT being a field theory, it's not at all clear that you're measuring the position of a single particle, because in field terms you're just measuring a statistic of the field in two different places in space-time (constructing an ensemble of data points or some other way of relating probabilities to statistics needs some care in this conceptual background, of course). Even if we think in terms of QFT being a particle theory, which IMO, and in the opinion of an increasing number of papers in the literature, goes somewhat against the current of the formalism, the two measurements may be measurements of the positions of different particles, not of two positions of the same particle.
I would further reduce Marek's comment that you don't need to tell when there is a collapse. If you're an instrument-level experimentalist, aka a lab technician, all you care about is that there is an analogue signal. You might notice that the signal has more-or-less discrete transitions between different levels or different types of behavior, and because of that you might start to record when those discrete events occur. Only if you care about theory, and want to give an explanation for the discrete events, do you mention collapse (although "collapse" is more an encoded description of the observed discrete structure than an explanation). I think it's only if you care about theory that you have particular wishes as to what features of the analogue signal you will convert to logbook or digital representation or put into a [published paper, which we can suppose will include particular interest in any identifiable discrete features there might be in the analogue signal.
Bjorn's Answer appeared while I wrote this. It's a good way to start discussing more detailed models of what measurements do (it's Useful!) in quantum mechanical terms. I can get carried away by QFT ways of thinking. In the QM sort of vein, you might try the book "Operational Quantum Physics" by Paul Busch, Marian Grabowski, and Pekka J. Lahti (don't be misled by the "Operational", it's far more theory than experimental, and also not elementary).