It looks like you are trying to find whether an item is magical at all and not what a magical item does. I also don't see the line about handling giving away that an item is magical. Understood that you are trying to run by the book but that rule is easily imported and has a whole slew of interpretations (just google your question for 5e and there's threads and threads about how to manifest that line in game). Otherwise, RAW, magical items typically require a fine or masterwork quality mundane item to make, that means you can easily describe the item as higher quality, depending on the rarity. Many of those answers call out that in addition to being a well made item, the magical enchantments keep the item unnaturally preserved, wchich can be at least a flavor thing to add if you don't want to outright tell players that an item is magical because it isn't in the book.
I don't believe you're looking for it but the rules to determine the qualities of an item are pretty explicit and found on page 207 of TNT:
"A creature can spend a short rest inspecting a magic
item, making an ability check at the end against
a DC based on the magic item’s rarity (see Table:
Identifying Magic Items) after searching its memo-
ries for references as it scrutinizes the magic item
for clues. The type of the ability check and any
skills used for it are at the Narrator’s discretion,
determined by the magic item and its origins, but
often include Arcana, Culture, History, Nature, or
Religion. On a success, at the end of the short rest
the creature recognizes what the magic item is and
remembers any command words it might require.
Whether or not a magic item is cursed requires a
success by 10 or more."
