The burn in period is the number of samples you wait before you regard the sampler as having converged - for the Markov chain mixing time to elapse. If it has converged so that you are sampling from the stationary distribution, then your samples should be from the desired joint distribution.
So then the sampling period refers to the number of iterates for which you then collect data from the sampler.
However you should note that terminology is not always universal -- some people may use somewhat different names for some of these things.
It's generally not necessary to 'skip' observations; in many cases it's much more efficient to keep the correlated samples and deal with the correlation. Nevertheless many people do skip ... but the size of the skips is not the sampling period. It's also not the same as the burn-in.
The skips are used to space observations so that the dependence between successive samples has time to die down - very much in the way autocorrelation dies down for an AR. This is quite different from the mixing time for the Markov chain. We don't need to wait for it to remix, just for the dependence in successive iterates to die out.
The purpose is quite different and the time required is usually different (generally smaller).
So you might - for example - burn in for 100, sample for 10000, and then take say every tenth observation, getting 1000 nearly-independent values. Or you might instead keep the whole sampling period for a shorter run, but then you must deal with the dependence in the draws from the stationary distribution. For example, variances of means will need to deal with the autocorrelation structure in the draws.
Your suggested scheme of making the skip interval the same as the burn in would only work if the burn in was longer than the necessary skip time. Since burn-in would usually be much longer than the necessary skip time, it would work but be incredibly wasteful of iterates. (If you're going to do that you might as well restart the sampler each time you sample one value.)