'Stative / Dynamic'

I

"Have" is a verb that has both stative and dynamic meanings, all well established; in this particular case of use with "meeting" "have" means "to do".

(OALD) 16 ​ have something to perform a particular action
♦ I had a swim to cool down.
♦ (British English) to have a wash/shower/bath
♦ We had a very interesting discussion about climate change.

In the case of "appointment" it is more difficult to make out the verb. From OALD or Cambridge Dictionary we see that an appointment is "a formal arrangement to meet or visit somebody at a particular time". There is however another meaning of the word appointment, and it can be inferred from the common phrases "have an appointment tomorrow" and "go to an appointment"; it is evident that the arrangement is not situated tomorrow but that it exists before that time and that is is something quite abstract; similarly people do not "go to the arrangement", which is utter nonsense. It follows that "appointment" has one more signification (apparently not taken into account by some dictionaries); this is that signification that is proper for verbal expressions such as "to go to", and "have" where "have" has a particular sense indicated by the continuous tense—"to have an arrangement" in the sense of being in possession of directives (arrangement) is without problem, well defined. Then, if we devise a supplementary sense of "appointment" as "the getting together with one or several persons at a place and time that have been decided through mutual arrangement". It follows that the verb "have" as, again, "to perform a particular action" will do and the continuous tense is without problem.

II

Certain explanations about this unusual usage of the continuous can be found in CoGEL § 4.28, p.200, and also in another source. (: unacceptable; ?: tending to unacceptability, but not fully unacceptable; ?: native speakers unsure about unacceptability)

Stative types A and B: qualities and states

Among stative situation types, a rough distinction may be drawn between QUALITIES […] and STATES […]. Qualities are relatively permanent and inalienable properties of the subject referent. The primary verbs be and have are preeminently quality-introducing verbs; but they can also introduce the less permanent situation types called states.
Contrast:

|QUALITIES|STATES| |:-------|:---| |Mary is Canadian.     [1]|Mary is tired.      [3]| |Mary has blue eyes.      [2]|Mary has a bad cold.       [4]| Normally such stative situation types do not occur with the progressive (this is especially true of qualities):

|QUALITIES|STATES| |:-------|:---| |*Mary is being a Canadian.     [1a]|?*Mary is being tired.      [3a]| |*Mary is having blue eyes.      [2a]|?Mary is having has a bad cold.       [4]| If sentences such as [1a-4a] do occur with the progressive, it is a sign that they have been in some sense reinterpreted as containing a dynamic predication. For example, Peter is being awkward signifies that 'awkwardness' is a form of behaviour or activity, not a permanent trait. If sentence [3a] were to occur, it would signify that Mary was pretending to be tired (ie indulging in a deceptive activity), rather than in a state of real lassitude. Although verbs with stative meaning have sometimes been called 'nonprogressive', we should observe that the definition of stative verbs is not so much that they are incompatible with the progressive, as that when they are combined with the progressive, some change of interpretation other than the addition of the 'temporary' meaning of the progressive aspect is required. This change of interpretation can usually be explained as a transfer, or reclassification of the verb as dynamic, eg as having a meaning of process or agentivity. The representative stative verbs be, hope, and resemble, are illustrated in Table 4.28:

Table 4.28

normal nonprogressive	nonnormal progressive	special effect of progressive
The neighbours are friendly.	The neighbours are being friendly.	Suggests that 'friendliness' is a form of behaviour (perhaps insincere).
I hope you will come.	I am hoping you will come.	Makes the speaker's attitude more tentative and perhaps more polite […]
Tina resembles her sister.	Tina is resembling her	With the comparative construction, the progressive turns the stative meaning into a process meaning[…].

Another characteristic of such verbs, when they are expressing states, is that they can be used in a 'temporary' sense in the simple present tense:

     The manager is away this afternoon. [5]
     The manager works this afternoon. [6]

For example, [5] is nonprogressive, and yet clearly refers to a limited period; in contrast, [6] cannot refer to a temporary state of affairs; instead, it has to be interpreted in a habitual sense ['The manager works this afternoon (being Tuesday) every week']. Equally, in [7]:

     The music they are playing sounds like Mahler. [7]

the choice of the progressive with the dynamic verb play indicates the same limited time period as the use of the nonprogressive with the stative verb sound.

_{Note
The constraints of the progressive cannot, it seems, be explained entirely in terms of meaning. Since the use of the progressive aspect has been undergoing grammatical extension over the past few hundred years, it is likely that its use is still changing at the present day, and that its description at any one time cannot be totally systematic. This would explain the difficulties faced by those attempting to account in every respect for the conditions for the use of the progressive in terms of semantic generalizations.}

Additionally, in Practical English Usage, second edition, p. 464, § 451 No 3, a note gives a general explanation, which is in keeping with user Edwin Ashworth's comment.

Note also that many 'non-progressive' verbs are occasionally used in progressive forms in order to emphasize the idea of change or development.
     These days people are preferring to take early retirement.
     The water 's tasting better today.