Confusion in this topic is understandable. I remember having had a hard time when I had to study the classification of such verbs according to different authors/grammarians.
Quirk et al., for example, consider the following verbs modals: can, could, may, might, shall, should, will, would, must, ought to, used to, need, dare.
Hugh Gettin and David Brazil offer similar classifications.
Eckersley & Eckersley call modals "special finites". They say there are twelve of them (be, have, can, do, shall, will, may, must, need, ought to, dare, used to). They also say that twenty-four is also a possible number if we consider "other derived forms (excluding the imperative) for tense, number and person, viz. am, is, are, was, were; has, had; does, did; should; would; could; might.
I personally find this classification a bit messy.
Celce-Murcia & Larsen-Freeman speak of "modals" (can, could, will, shall, must, should, ought to, would, may, might) and "phrasal modals" (be able to, be going to, be about to, have to, have got to, be to, used to, be allowed to, be permitted to).
Some authors (Stockwell, I think, among them) speak of "modals" and "semi-modals".
R. A. Jacobs (tihs is the one I prefer) speaks of "core modals" (may, might, can, could, will, would, shall, should, must, ought to, need, dare) and "periphrastic modals" (multiword verb idioms). In the category of periphrastic modals he includes forms such as be likely to, be due to, be apt to, be willing to, be unable to, besides the ones mention by Celce-Murcia as phrasal modals. He does mention had better and would rather as being "similar" to core modals; yet he doesn't say they *are* modals.
So, I guess it all comes down to what author you follow -if you choose to follow one in particular. Probably this is no big deal really, it's just that I'm always paying attention to tiny details... perhaps too much attention. Don't let me bother you with this any longer.
The day *I* write a grammar book, I'll call things anything I like! ~chuckles~
Miriam