First of all, of course peaks in NMR are counted by their "tops" - that is most of the time - when spectra are presented in the absorption mode. Simple 1D spectra are.

Secondly patterns like doublet, triplet, etc. have characteristic parameters: (1) obviously - number of peaks, (2) coupling constant - all immediately ajdacent components must be separated by the same frequency interval called J-coupling, and (3) - integral ratio of components. The latter is 1:1 in doublet, 1:2:1 in triplet, 1:3:3:1 in quartet, etc. The ratio is determined by a so-called "Pascal triangle".

Clean patterns like these are observed only when value of J coupling is much smaller than separation between resonance frequencies of the coupled nuclei. In practice this separation must be >10-fold of J coupling. If this ratio is smaller, simulation of NMR spectrum will be required to determine true values of the J coupling. I guess in such cases you'll be compelled to use label "m".

Things get more complicated when peaks overlap and when there are composite splitting patterns, like doublet of doublets, doublet of triplets, etc. That's when you really have to start scratching your head, alternatively you can analyze a 2D DQF COSY or some other appropriate experiment. Maybe someone else can fill in on this.

Please take a look at the two papers cited below. They go in depth through a general method of extracting multiplet patterns from NMR spectra.

1. A Practical Guide to First-Order Multiplet Analysis in 1H NMR Spectroscopy by Hoye, Hanson and Vyvyan
2. A Method for Easily Determining Coupling Constant Values: An Addendum to “A Practical Guide to First-Order Multiplet Analysis in 1H NMR Spectroscopy” by Hoye and Zhao (2002)