Tag is the secret word.
Determine frequency of letters across all 6 words. Then assign a letter count to each word. For example, cat is 142 because c appears once, a appears 4 times, and t appears 2 times.
So I get
Observe that cat and max have the same frequencies (except for a permutation). So if, for instance, one player had the letter c and concluded the word was cat, the other two players can't tell if the word was cat or max. Same for dog and dim. So I can eliminate these 4 words.
That leaves has-141 and tag-242. If the three players followed my thinking and one had the letter h or s that player would immediately hand in his paper, followed instantly by the other, and the third would be close behind. Since the problem says (implies) that some thought and time went by before the first player handed in his paper, this means that that player took the extra time after eliminating the first 4 words to also eliminate has since no one had immediately jumped up. Similarly for the second player. These two players must have had the letters t and g. The third player had the letter a and had to wait for the first two to act. I conclude that tag is the secret word.