As I recall, Appraiser exp is granted to whoever first shows up at the depot with the job trade at depot. (not bring item to depot.)
There's a reason one usually makes sure to add at least one appraiser point to the starting seven.
That makes one bit of exp per caravan. not exactly an easy thing to bring up.
The first is (mostly) right. The first dwarf that trades anything gets some appraisal xp, usually enough to bump him/her up to 1
According to my experience (which consists of _never ever_ assigning appraisal skill at embark), appraiser experience is gained by the act of opening the trade window for the first time. If the caravan in question is the dwarven autumn caravan and it has brought wagons, this is generally enough to give the dwarf in question adequate or better appraisal skill.
The simple trick is to either pick a dwarf whom you consider fitting for broker or call a trader with the "anybody may trade" setting at the depot and make the dwarf who shows up the broker. The experience gained by opening the trade window will give enough experience to have all values displayed - if the dwarf gaining that experience is the broker.
You (o)ffered the goods, which is a gift, not a trade
Since i always get values displayed the moment i open the trade window (with a dwarf without prior appraiser skill), offering vs. trading shouldn't be relevant.
Traders may get more experience and/or experience in other skills per successful trade, i don't know about that. But the big decisive source of trading experience is just the act of getting trading goods displayed, and that only seems to work once per caravan (or per item shown in the trade window).