this has sources cited regarding it:
http://homepages.paradise.net.nz/rochelle.f/The-Discovery-of-steam-power.htmlSee also Heron Alexandrinus's works Automata and Pneumatica.
Even wikipedia names the aeolipile as the first steam turbine engine (to be contrasted with simply using steam for power, which Heron's books above cover different uses thereof).
One might conclude the only reasons further development of the concept ceased are either economical or because the knowledge was lost during the dark ages.
To get a functional steam engine all you have to do is add a shaft to the spinning object on the aeolipile so the rotational power can be exported. This falls into the category of 'immediately obvious', especially as archimedes screw was a well-known device at the time, and thus anyone with any mechanical knowledge would have been familiar with the translation of rotational energy into non-rotational energy and vice-versa. So yes, the aeolipile certainly counts, and its called a *steam turbine engine* in many sources, both scholarly and public.
(Regarding economics - slaves were pretty cheap in the days of the Roman empire.)