A lot of the answers depend upon what time period, geographical area, and ethnic culture you are interested in.
First, throwing a hawk is a SPORT. Nothing more. It's competition for fun and/or profit. Any "hawk" can be thrown, but some work better. And the "hard core" competitors choose, tweak, and treat their throwing hawk like the precision sports equipment they view it as. Many events have had to write up pretty detailed rules for hawk throwing - because of the little tricks and gimmicks that some hard-core win-at-any-cost competitors have brought to what had been friendly competition. But once rules are written up, there are always those people who spend more time trying to get around or bend the rules than accept and work within them.
And so much of throwing a hawk comes down to personal preferences. Short handle, long handle, light head, heavy head, wide/long cutting edge, smooth handle or grooved for gripping, etc.
A hawk with a hammer pole does help increase its usefulness as a WORKING TOOL - pounding in tent stakes being a big use. But it also narrows down the what/when/where of its' style, shape, and who would have used it. Plus, that pole pushes the time period for any hawk or axe mostly into the late 1700's, but more generally into the 1800's. The round eye hawk/axe with no hammer pole goes from the 1800's (even 1900's) all the way back well into the Middle Ages and earlier.
In the end, it's all a matter of personal choice. How well do you wish it to fit into your chosen time period, culture, geographic region, and how you expect to use it. And that final concern - the LOOK you want in a hawk - which often ignores all the other considerations.
Just my humble thoughts and opinions to share. Take them as such.
Mikey - out in the Hinterlands
p.s. Be warned! Sharp toys can quickly become ADDICTIVE! Once you get one, the next just seems to ... fall into your hands as if fate had guided it! After that second one, all is lost - and only strong will, liquid therapy, and constant supervision of the spouse will control the addiction.