(Disclosure - I am part of the admissions staff at a linguistics department in a UK university)
Talk to the department you are applying to first. Although they won't be able to give a definitive 'no - you won't get the PhD offer if you apply', most departments (at least in the UK) will be happy to talk, informally, about whether you fulfil the prerequisites.
There are some general points to remember though. A student enrolled on a PhD will be required to submit a substantial body of scientifically original work at the end of the course, and your ability to deliver this is what the admission committee will be trying to gauge. In your case, it is worth remembering that linguistics is the scientific study of language* - and actually has a very little to do with language skills themselves. Unfortunately, knowing Chinese and Spanish, and being able to teach English as a second language, do not provide evidence of your understanding of the science of linguistics (although they clearly show your enthusiasm with languages, which will be a plus). You have gone about correcting this in the right way, by doing MOOC and phonetics courses, but this may not be enough. The committee have to be sure you will be able to write up a good scientific thesis - if you are accepted, and are subsequently unable to write a good scientific thesis, (in the UK at least) they may stand to loose funding.
You don't say where in the world you are applying, but if it is in the UK, most prospective PhD's students have decided beforehand - at least vaguely - the broad sub-field they plan to do their PhD in, so they can approach a PhD supervisor at the department they intend to join. (So, for linguistics, subfields include semantics, first language acquisition, phonetics, prosody etc). This is why masters courses are so useful as a first step before a PhD - they allow the student to be sure of the subfield they want to work in, to work out if that department has the supervisor they want, and to familiarise themselves with the field so they can locate potential supervisors in other universities.
You say you aspire to be a professor one day - if so, perhaps a masters course would help you find that subfield of linguistics you really enjoy, and to find a supervisor that will inspire you to do great science.
EDIT: Added after further information by the original poster
I had actually considered asking but was worried about making a poor
first impression. I don't want to start out appearing doubtful of my
abilities to do graduate study.
If you are asking for information about the course, or clarification about entry requirements, this will not be seen as you doubting your ability. It will be seen as you having the maturity to find and clarify the information you need to make the right decision about which course is right for you.
I plan to apply to UC Davis - who exactly should I ask, do you think,
about their general admission standards for MA vs PhD?
The ‘graduate administrator’. For UC Davis’ graduate linguistics programs, the name and email address of their administrator can be found at the bottom of this page: http://linguistics.ucdavis.edu/graduate-program/admission (her title is ‘graduate co-coordinator’). If you email her, then she will answer any questions you have herself or forward it on your behalf to whoever is best placed to answer them.
I agree that (to me at least) the prerequisites to this course seem a little ambiguous, saying that, if you haven’t taken some of their prerequisite undergraduate courses, you can take them during the first year – which makes it seem like they are not prerequisites at all. But this is a perfectly sensible clarification to ask by email, something along the lines of:
I am considering applying to your department for a graduate course and I am trying to decide whether your masters or PhD course is more suitable for my circumstances.
I see that the prerequisites for the PhD on your website are courses equivalent to four undergraduate classes.
I have taken the online [fill in MOOC course here] linguistics course, which covered [X] hours of teaching and [X] hours of course work. Would this set of courses fulfil the prerequisites for your PhD program?
As I say, this is a very reasonable question, and the email itself will not negatively affect your chances of getting in. There are two further pieces of advice I would give based on the information you have given:
Each course they list as a prerequisite is 3 or 4 hours each. Some include course-work or discussions. I'm afraid that, if your MOOC course was not of a similar high standard, you should be prepared for the answer to be ‘no’.
You mention that you completed a TESOL certificate. While this is a great thing to put in your application to show your enthusiasm for the subject area of second language development, I suspect the admissions committee may not feel it demonstrates your ability as a scientist in this subfield. But again, ask the department first to make sure - they may feel differently.