[Goldenwolf]

Is a person a pyromaniac if they feel the impulse to start or see fire to feel happiness or relief afterwards?

[Alty]

Here's what I found;

Deliberate and purposeful fire setting on more than one occasion.
Tension or affective arousal before the act.

Fascination with, interest in, curiosity about, or attraction to fire and its situational contexts (e.g. paraphernalia, uses, consequences).

Pleasure, gratification, or relief when setting fires, or when witnessing or participating in their aftermath.

The fire setting is not done for monetary gain, as an expression of sociopolitical ideology, to conceal criminal activity, to express anger or vengeance, to improve one's living circumstances, in response to a delusion or hallucination, or as a result of impaired judgment (e.g. in dementia, Mental Retardation, Substance Intoxication).

The fire setting is not better accounted for by Conduct Disorder, a Manic Episode, or Antisocial Personality Disorder.

Criteria summarized from:
American Psychiatric Association. (1994). Diagnostic and statistical manual of mental disorders, fourth edition. Washington, DC: American Psychiatric Association.