There actually is a definition of an "activity engaged in not for profit", or more precisely a set of relevant factors, and intent does not override those factors in any way. As the Regs specifically state:
In determining whether an activity is engaged in for profit, greater weight is given to objective facts than to the taxpayer's mere statement of his intent. Treas. Reg. §1.183-2(a)
The relevant factors from the Regs are:
(1) Manner in which the taxpayer carries on the activity.
(2) The expertise of the taxpayer or his advisors.
(3) The time and effort expended by the taxpayer in carrying on the activity.
(4) Expectation that assets used in activity may appreciate in value
(5) The success of the taxpayer in carrying on other similar or dissimilar activities.
(6) The taxpayer's history of income or losses with respect to the activity.
(7) The amount of occasional profits, if any, which are earned.
(8) The financial status of the taxpayer.
(9) Elements of personal pleasure or recreation.
