For completeness, I should mention that there are (at least) three widely used definitions of BEC. The budding physics student should probably not worry too much about the differences, but they do matter.
1. Order via broken U(1) symmetry. Usually signalled by the existence of a non-zero <c>. Downside: can't actually happen in any finite system.
2. Off-diagonal long-range order. The existence of <c*(r) c(0)> as r goes to infinity. Again, not really appropriate for any system where the limit can't actually be taken (most atomic BECs are pretty small).
3. The third is somewhat more technical: the existence of a "macroscopic" eigenvalue to the reduced density matrix <c*(r) c(0)>. Really, it's just saying that there is some eigenstate in the density matrix which is dominant over the others. Usually, the ratio will be something like 1:N where N is the number of particles. One can obviously have more marginal cases (1:100 or less). Neverthless, it actually includes the previous two as special cases.
The first definition is probably the most widely used, usually taught starting from advanced undergraduate level. In that definition, a BEC requires a broken symmetry order, which as a matter of principle is not possible in 2D or 1D because thermal fluctuations would destroy it at any non-zero temperature.
I'm not a polariton expert, but my understanding is that the transition seen is actually of a Kosterlitz-Thouless kind, rather than the usual textbook BEC. I personally do not think these two transitions are the same, essentially because one is topological and one actually involves a more classical symmetry change.
