Solution 1:

It appears that this is a British/American distinction. The Corpus of Contemporary American English (COCA) reports 92 incidences of “fill out a/the/this form” and just 2 of “fill in a/the/this form”, clearly establishing “fill out” as the standard idiom in American English. I haven’t worked out how to search the British National Corpus yet, but I wouldn’t be surprised if the results were reversed there.

Edit:

OK, I got the BNC to respond to queries—although it sure takes its sweet time—and I got 19+7=26 results for “fill in a/the form”, and 5+1=6 for “fill out a/the form”. So it does appear that British English favors fill in over fill out, although not to the degree to which American English favors fill out over fill in.

Solution 2:

Both are perfectly acceptable.

Solution 3:

As an Englishman living in the US for almost 20 years, "fill out" still sounds jarring to my ears. I had never heard it used before I came to America.

I rarely hear "fill in" on this side of the Atlantic.

Solution 4:

In my dialect of American English, you "fill out the form" by "filling in the blanks" on the form.

P.S. to fill out the form is to complete it. To fill in the form is to supply information as required.