Never had a problem with slippage so far, although most of the larger bits I use are hex shank.
I always just turn the chuck part by hand to tighten it down; the belts & motor usually have enough resistance to get it 90% of the way there and then you just hold the top/bottom parts and give it one last twist. Removing the bit is similar, actually easier...just rotate the spindle/chuck by hand in the forward direction, then stop it suddenly - inertia from the motor and pulleys will usually loosen it up right away.
If I had to take it out, I could probably get a drift punch or something similar in the slot on the side of the spindle...or you could likely use a different arbor, like the one in your first link. It's all standard tapers.
Definitely way more accurate (especially with smaller bits) and 'tight' than the junky one that came with the HF press.