Question: I am a Master's student in math who is doing pretty well so far. 
I aced my first two classes (real analysis) 
and my summer session in probability theory is going well, too. 
Even before my Master's study, I already thought of a PhD. 
Nonetheless, there are concerns:

1. My good grades were the result of grit, not brain, 
meaning that I had to spend time to understand what the book is telling me. 
Furthermore, a lot of times during exam and homework, 
I had to try and fail a few times before arriving at the solution.
2. I did not study math during undergrad. Before my Master's study,
I gulped down three semesters of calculus, plus LA and DE in one year. 
So compared to other students, I am already behind. 
In the next two months, I will learn complex numbers on my own
and review my linear algebra.
3. I am a professional in a field that has nothing to do with math or research. 
After the holiday, I will speak to my advisor about taking a thesis class. 
It's not an insurmountable obstacle, 
but, in an unrelated field, I do have less time to concentrate on math.

At the end, how big is the jump from Master's to Phd? 
Especially after what I said in (1), I do worry that it is beyond my ability. 
I do not hold anyone responsible for my decision. 
So, please kindly offer your best assessment of my situation.

Answer: 
There are a few different things going on in this question, I'll try and address them separately.

First of all, heed JeffE's words about the impostor syndrome. The idea that some people are effortly geniuses (which is inevitably followed by "but not me") is untrue and distinctly unhelpful.

Moreover trying and failing a few (or many!) times before coming to a solution is the very *definition* of doing research! The important question you should ask yourself is if you enjoy the challenge of struggling against a problem; if so, then research would be a great fit.

As for your background, you seem to have a plan to address the gaps in your background, and if you're doing well in your master's programme, I wouldn't worry too much about being behind the other students. Another challenge may be if you have a full-time job; doing a Ph.D. with a fulltime job is by no means impossible, but requires a some great time management skills. Anyway there's plenty of resources on this forum to help with this issue.

Finally, as for the question in the title, I would say it depends the most on what you plan to do with your Ph.D. after you graduate. I'm a recent graduate (in math) from a well-known North American school, and the quality of the Ph.D's varied greatly. I think it really depends on what you want to do with the degree. If getting a Ph.D. is the farthest one wants to go in academia, the standards for graduating are pretty low, and I think can be achieved not too strenuously. If one wants to be competitive in the academic research job market, the bar is quite a bit higher.
As Austin Mohr, pointed out, it is unlikely a stranger on the net fully answer this for you, but what you are experiencing is similar to what I experienced, so I can tell you how it was from my perspective.

Given what JeffE commented, studying through true 'grit', does involve 'brains' as you are developing and applying problem solving skills and stategies - which has clearly been effective, as you mention, you have good grades. So, it seems you have developed effective study techniques.

Point 2 is pretty much not an issue, based on the skills and strategies you developed in point 1.

If you do an unrelated thesis, as you mentioned in point 3 - look at this way, you'll be developing the research and synthesis skills needed for a PhD.

What I found about the transition between the Masters and PhD, was that it wasn't so much of a leap, but a case of using the skills developed a lot more thoroughly.

I hope this helps.