This depends on what you are asking, it is akin to wanting to know if you desire a weighted average (weighted by the # of days/trip) or a simple average.
It is possible, and even likely, that the cost of a trip for a single day differs significantly from the daily cost of a multi-day trip: people who are in Italy for a single day or two are perhaps more likely to pack their day with multiple activities and expenses, whereas folks with several weeks may choose a small # of things to see each day. I believe this would argue against a weighted average, because we do not want to inflate the effect of longer trips since it is likely that their behavior is independently different from the behavior of shorter trips. I would report an "average daily cost" by taking the latter approach you outlined.
As an example, if I have 5 people who took trips with the final details below:
1) 20 days, \$600 total, \$30/day
2) 2 days, \$200 total, \$100/day
3) 1 day, \$100 total, \$100/day
4) 3 days, \$240 total, \$80/day
5) 2 days, \$180 total, \$90/day
Approach A: The average amount spent for all days spent in Italy was: (600+200+100+240+180)/(28) = \$47.14/day
Approach B: The average daily amount spent per trip in Italy was: (30+100+100+80+90)/5 = \$80/day
Approach A is misleading, of all 5 trips people took to Italy, only one had a trip as cheap as \$47.14/day. Meanwhile, the vast majority of people who took trips to Italy should expect to spend at least \$80/day, which is closer to the average in
You should also see the inspection paradox