What kind of verb classification is "causative"?

Causative is a term with a lot of uses, in several different contexts. First, as usual, we start out correcting misimpressions. Let and allow are not exactly "causative verbs" -- rather, they participate in some causative constructions, as does have (as in I had my tires rotated). This does not make them causative verbs, however.

Verbs, and other predicates, can describe either events (Active predicates) or states (Stative predicates). Either events or states can take a long time or a short time; some occur at a point in time, others spread out over a long period. There can be changes -- growth or loss, age, etc. -- and these can be sudden or gradual. Events and states can start or stop suddenly or slowly. There are predicates that cover all of these possibilities, and more.

Causative verbs, like kill, darken, bring, wake, and inflate, all have the sense of
Cause to Become for states, and Cause to Happen for events.

• Frankie killed Johnnie means 'Frankie caused Johnnie to die'
• The paint darkens the room means 'The paint causes the room to become dark(er)'
• Mary brought John to the party means 'Mary caused John to arrive at the party'
• They woke me early today means 'They caused me to become awake early today'
• She inflated the balloon properly means 'She caused the balloon to become properly inflated'

Like Positive, Comparative, and Superlative (big, bigger, biggest, the three degrees of comparison), Causative comes as part of a triad, in fact the top of it.

Both active and stative predicates can have beginnings and endings -- changes of state. Predicates that refer to change of state are called Inchoative (pronounced /ɪn ko ə tɪv/). Begin, start, finish, stop, end, and continue are basic inchoative verbs. But practically any predicate has an inchoative and a causative counterpart, if there is any need to refer to change or causes of change -- and there usually is.

Sometimes these three predicates are different words, like

• stative dead, inchoative die, causative kill (often the stative is a predicate adjective)

but more often they are simply different forms, nominal, adjective, or verbal, of the same root:

• broken, break, break (often the causative and inchoative have the same shape)
• have (as 'possess'), get (as 'receive'), get (as 'retrieve')

Get, in particular, is a very handy verb; it can be used with practically any predicate adjective (and as get to be, with practically any predicate noun; the to be is optional with predicate adjectives) to indicate either inchoative 'become' or causative 'cause/make'.

• He was tired/active/writing/kissed today. (states and events)
• He got tired/active/writing/kissed today. (states and events becoming)
• Indef got him tired/active/writing/kissed today. (states and events being caused)

Stative/Active - Inchoative - Causative is just one set of many characteristic types of predicate.