Those are different things though. The hunting log refers to a range of enemies that you prove that you can defeat, a trial if you must. By fully completing it you get your reward, "Master Pugilist's Ring" to mention one, which defines you as a "Master of X" of that class.
The daily/weekly hunt marks are different, they are marks wanted by the Allied Hunt Clan and the Ishgardian Centurio Clan, wanted criminals so to say, and you get payed with a currency after hunting them down. They are not a registry of enemies or trial. Nothing to do with class related things, hence you can find them in you key items as bills.

I do agree with you though that a bestiary would be nice, a full detailed registry of all ingame monsters, like in all classic FFs. But I don't see that coming, it would never be complete to mention one thing.