Short answer: according to EU legislation, a stick deodorant can reasonably be included in the ‘liquids, aerosols and gels’ category.
I think that the relevant legislation in this case is the Commission Implementing Regulation (EU) No 2015/1998 of 5 November 2015 laying down detailed measures for the implementation of the common basic standards on aviation security, which states:
4.0.4 For the purpose of this Annex:
‘liquids, aerosols and gels’ (LAGs) shall include pastes, lotions, liquid/solid mixtures and the contents of pressurised containers, such as toothpaste, hair gel, drinks, soups, syrups, perfume, shaving foam and other items with similar consistencies;
‘Stick deodorant’ isn’t specifically mentioned, so it comes down to an argument as to whether it can be included in one of those categories. I think it could be reasonably claimed that a stick deodorant is a ‘paste’: the Oxford English Dictionary entry for ‘paste’ includes, amongst other senses, ‘Any moist but fairly stiff mixture’, which certainly describes all the stick deodorants I've come across. This particular stick deodorant can also be included under ‘liquid/solid mixtures’: the ingredients of ‘Lady Speed Stick’ include:
elaeis guineensis (palm) kernel oil, stearyl alcohol, cyclomethicone, C12-15 alkyl benzoate, PPG-14 butyl ether, hydrogenated castor oil, hydrogenated soybean oil, PEG-8 distearate, fragrance, hydrolyzed corn starch, behenyl alcohol
Of these, at least cyclomethicone and PPG-14 Butyl ether are liquid at room temperature.
In practice, enforcement seems to vary (as with so many aspects of airport security): this flyertalk thread has a several anecdotes of stick deodorants getting through security, and several others of them not getting through. The bottom line seems to be that it’s a bit of a borderline case, but if the security staff decide that it’s a ‘liquid, aerosol, or gel’ then there’s no solid legal basis on which to dispute this.
(As a side note, it doesn’t seem entirely clear that the item under dispute was a stick deodorant: the airport police and Washington Post both describe it as a ‘roll-on’. But then, that’s far from the only point where accounts of the incident diverge.)