Merely a question of personal taste. The advantage of the strap is that you won't lose your weapon should you temporarily lose your grip, the advantage of not having any is that you can throw the kama (a number of schools teach such techniques). Unfortunately I studied nichogama but extremely shortly, so I'm sure some more competent members should be able to give you some more detailed information. I will just say that if I were you, I would focus on the pins holding the blade in place more than the strap. Two pins are HIGHLY preferable than one only. I don't how what to think about those kama you pointed out since that red chrome thing prevents me from seiing that clearly. I will just say that they look way too flashy to my taste (also not as much so as some I've seen
, but it's just that: merely a personal taste. I clearly prefer a good ol' traditional looking set, but hey, to each their own...
One last thing: the first kanji on the handle is a bit too blurred on the photo for me to identify it, but the other two are bu jutsu
which simply mean "martial arts".
Hope it helped!