There are about two different questions here. One is, "what are the phonetic properties for classifying the English consonants?". In order to answer the question, you need a theory of phonetic descriptions. The second is, "what are the phonetic labels associated with the IPA letters?". I will start by answering the second question: here is the IPA chart. IPA is an official standard, and that is a link to an official IPA chart. From this, you can see that the main table of consonants is labeled "pulmonic" (there are also clicks, implosives and ejectives which are labeled non-pulmonic). Other consonants are not labeled one way or the other, and that is where you will find "w". Although "w" is pulmonic in the ordinary (scientific) meaning of the term, it is not specifically labeled "pulmonic". There may be no historical record of why the association did not apply the term "pulmonic" to the set of "other symbols".
If it is not clear, Wikipedia is a bad resource if you are interested in official and standard IPA: they throw terminology around ad libitum. The term "coronal" is not used by the IPA. The voiced epiglottal fricative ʢ is not part of the main set labeled in IPA as "pulmonic", it is in the "other" set along with "w". The column that Wiki calls "palato-alveolar" corresponds to IPA "postalveolar", and is called "alveopalatal" in standard linguistic usage. The Wiki descriptive labels are a highly variable merger of many trends in linguistics, standing sort of midway between IPA and feature theory (closer to IPA than to feature theory).
There is a standard phonetic feature set (20 features) for describing language sounds, set forth in Sound Pattern of English, which is the source of terms like "coronal", but "ʃ" would be [+coronal,-anterior], that is, it decomposes sounds into atomic properties, whereas we normally speak in more "molecular" terms (like "uvular", which is the result of 3 feature setting). There is a somewhat simple answer to your question based on that system, since using the letter "p" strictly according to IPA norms, we can specify its phonetic properties ([+consonantal,-syllabic,-sonorant,-voice,-coronal,+anterior...]). Each possible sound is an integer, each bit position represents + or - that Nth feature, all you have to do is set a standard order for features. For example, if bit 5 is "voice" then "s,θ,h..." have bit 5 off and "z,ð,ɦ..." on: and that translates trivially to "voiced" vs. "voiceless". Terms like "velar" are special cases, where bits 12-15 have the pattern "0010".
Assuming that you want to stick with descriptive terms like "velar", you have to pick some set of terms. The IPA has officially assigned terms to letters: "w" is a "Voiced labial-velar approximant", and "ʍ" is a "Voiceless labial-velar fricative". Compositionally, diacritics can be added to letters and corresponding to the description, where "w̥" would be a "voiceless (voiced) labial-velar approximant". This allows you to handle "n̩" which is a "syllabic alveolar nasal (pulmonic) consonant". The problem is that IPA gives official descriptions of letters, but not rules for merging descriptions. This is evident in the fact that "w" is defined as "voiced", the voiceless diacritic adds "voiceless", but then you get a clash between "voiced" (from "w") and "voiceless" (from the diacritic).
Notice that IPA calls "w" a labial-velar.
Features and phonetic definitions are given in The Sound Pattern of English: they hint at around 40 features but half of them received no uptake. The features that were adopted in the field were sonorant, syllabic, consonantal, coronal, anterior, high, low, back, round, distributed, nasal, lateral, continuant, delayed release, tense, voiced, strident, long, and stress: two others were proposed by Halle & Stevens shortly thereafter and adopted (spread glottis, constricted glottis), and one feature was renamed (Advanced Tongue Root, formerly "covered"). These features are still used in some textbooks, such as Introducing phonology.
More recently and inspired by "feature geometry", there has been movement in the direction of using traditional phonetic labels with a plus sign, for example "[+approximant]", but this trend has not standardized.