Most turboprops have relatively small engines which means that electric starters are generally the most appropriate. Therefore, the options available for the designers for a source of electric power are batteries, APU or GPU. In each case, the electric source has to be reliable because its failure may result a hot start and subsequent (expensive) engine damage. But this has to be balanced against weight (batteries and APU are heavy) and availability (a bush strip is unlikely to have a GPU). But I think I would be true in saying that most modern turboprop operators prefer their aircraft to be started by sources in the following order: GPU, APU and then battery. Only in a smaller turboprop would a battery start be the "preferred" option.
Your average jet engine however, is a fairly large beast and so far air (driving an "air motor" connected via an accessory gearbox to one of the shafts of the engine) has been the best method of starting it. Again weight vs reliablity vs cost will have been taken into account. Air is easy to pipe around the place, it's normally readily available from APU's (or ground Air Start units when your APU is broken), installed to supply electricity when the engine driven generators are not running and will allow a consistent, reliable start.