I'll have to add this to the questions, seems to be common.
So to answer your question, YES, you must have your Atma weapon equipped in order to complete the book entries. The lady who gives the books originally says you must have your Atma relic weapon equipped during these "Trials". Also, you can test this. Go to an area where you have to kill specific enemies for your book. Equip your Atma Weapon, and look above the enemies head, you should see a Dot with a spiked circle around it. Now take off your weapon, or switch jobs, the symbol will disappear, meaning you won't get credit if you kill it. Same can be said for the dungeons, leves, and fates.
Also your side note is very helpful.