I do not have the same edition of Carnie's textbook as you have, and mine has no tree featuring an 'active' V on its p. 305, but the distinction between the 'active' V and the 'normal' V(P) was introduced in late P&P and early Minimalist Theory as a means to represent in syntactic terms the internal semantic structure of verbs expressing events involving actions/activities of an Agent. Of course, as there are many verbs that do not describe events implying action or activity, the 'active' V head can be expected to not be represented in the syntactic trees corresponding to such non-agentive VPs.
The reason underlying the postulation of an abstract 'active V' head was an attempt to make syntactic structure bi-uniquely correspond to the internal semantic components detectable in the meaning of different classes of verbs and is perhaps clearest in the case of homophonous lexemes like open that may alternatively describe a) a mere state of affairs (x is open), b) a change of state of x (into that state of 'openness'), c) a change of state of x caused by a non-intentional causer, and d) a change of state of x intentionally caused by an agent, as in The door is open [state], The door opened [change of state], The wind opened the door [unintentionally caused change of state], and I opened the door [change of state intentionally caused by the action of an agent], respectively. An active sentence like I opened the door, in other words, describes an internally complex event in which the Agent (here = I) does something (= an action, the Causer) that causes the Theme (here = the door) to become (= a change of state) open (a resulting state of the door).
The way to transparently represent those covert 'semantic' components of I opened the door in X-bar syntactic terms in late P&PT/Minimalist Theory was to assign to its VP an X-bar structure containing no less than four hierarchically ordered VP shells, each with its head, complement and specifier, i.e., .... [__ 'V-DO' [__ 'V-CAUSE' [__ 'V-BECOME' [__ V-OPEN the door]]]], where [__ V-OPEN the door] is the complement of V-BECOME, [ __ V-BECOME [__ V-OPEN the door]] is the complement of V-CAUSE, and [__ V-CAUSE [__ V-BECOME__V-OPEN the door]]] is the complement of V-DO (= your 'active V').
Granted that underlying 'shell' structure in the VP, the 'active' meaning of 'open' in that sentence is accounted for by making the lexical head OPEN raise 'bottom up' from V head to V head (by 'head-movement') acquiring the successive meaning components (BECOME, CAUSE, DO) above it. On the other hand, the initially internal argument of OPEN (= 'the door') must successively raise into the Specs of 'V-OPEN' and '__ V-BECOME+OPEN', the V-CAUSE+BECOME+OPEN head must acquire a specifier of its own with the thematic role of Causer, and, finally, the V-DO+CAUSE+BECOME+OPEN head (= your 'active V') must assign the AGENT role to its own specifier (here = I).
A non-agentive sentence like The door opened, on the contrary, describes just a change of state (= 'The door became open') with no action or causation implied on the part of any Agent or Causer. Consequently, the 'upper' Action and Causation layers of event-structure projected in the case of the active sentence I opened the door will not be projected for the VP of The door opened.
If the event had been 'causative', but not 'agentive', as in The wind opened the door, the VP structure of that sentence would contain the CAUSE, BECOME and STATE 'shells', but not the ACTION shell at the top, nor an Agent argument, i.e., three, instead of four (or two) VP 'shells'.
Finally, the VP of a sentence like The door is open, which describes a mere state of affairs with no change of state, causation thereof, or intentional action on the part of an Agent implied, consists of only the lower 'stative' VP shell [__ V: open [the door]] and no head-movement (V to V) occurs. The only movement involved in that case is the raising of the internal argument of V-OPEN into Spec OPEN and then, successively, into higher specifiers until it lands in Spec Tense and checks its nominative Case feature there.
In sum, 'active V' must be expected to be present in, or absent from, VP structures depending on the 'event structure' of the head verb projected in each case. In VP's that describe states, processes (e.g., rain), changes of state, or unintentionally caused changes of state, 'active V' will not be projected; on the contrary, in VP structures describing actions or activities of an Agent, it will unless the sentence is 'passive' (= a 'state', usually expressed by the stative auxiliary verb be). In that case, the passive VP will, of course, contain no 'action' tier and its implied Agent argument will be either entirely suppressed or demoted to the status of a by+NP adjunct.