Contracted full time to part time without contract

Greetings,

I got a job at a cafe as a full-time worker with signed contracts, but before finishing my 3 months probation, I opted for part-time with my boss which she had agreed to. 

It has been more than a month since I started my "part-time" and also receive said adjusted part-time monthly salary. However, I just found out that my boss claims that as a part-timer, I'm not entitled to any annual leaves, and instead she would just count my leaves as unpaid leaves as she now pays me by the number of days worked in that month. 

My main questions for this situation are.

1. Should my pay be per day basis?
As far as I understand from using Google, that should be for a casual worker and not a part-timer. 

2. I signed the full-time contract, but have yet to sign any new contract for my part-time switch. According to my full-time contract, I need a month's notice before I can quit. But I have already been an "unofficial part-timer" for a month now. How would my termination be approached should I wish to quit? Would I still need a month's notice?

@ Klein

1.  You did not specify what your "part-time" is about. Is it suka datang, tak suka tak mau kerja? Those part-timers is still protected under employment laws E.A.1955 but must have specifics and a proper schedule /jobscope signed agreement /contract, otherwise is just daily or hourly paid "casual worker". It is different from WFM contract workers.

It is different from AUSSIES current laws on part-timers which defines them as the same as full time employees. In any case Malaysian laws is that those hourly paid part-timers is higher than full time employees (rates per hour), as such there is no annual leave accorded or even medical benefits or even any insurance coverage if a worker (part-time) died on the way to workplace or workplace accidental deaths and/or disablement.

2. also receive said adjusted part-time monthly salary.
The moment you have accepted wages in this manner, you have voided the full time employment agreement and without signing a new proper agreement, you can be subject to slavery employment which you reserves the right to walk away immediately. You have charted your own course.

As far as I understand from using Google, that should be for a casual worker and not a part-timer. 
There are daily paid, piece paid, hourly paid, weekly paid on both casual and part-time workers. Do not apply employment laws of other countries to be same as your own country. Be Malaysian. Google is never 100% correct.

Would I still need a month's notice?
You can walk away anytime after you receive your wages.
The company can also terminate you anytime oso.
There is no signed contract /agreement.