Hello again and thank you for your reply. In terms of future job applications, this really doesn't have to be an issue. What I mean by that, is that you simply tell your prospective employer what the facts were in your leaving your prior job, e.g. the job changed from a part time to a full time and I could only work part time at that moment, so we parted ways. Your old employer isn't going to give a prospective employer a lot of facts because they don't want to be exposed to a potential defamation of character suit. They would just confirm that you worked there and what you did.
The bigger issue is unemployment. That is a very fact specific and more complex determination. Generally speaking, an employer has the right to change the terms and conditions of employment and the employee, if they are "at will", which you would have been since you had no written term contract, must agree to those changes. If they refuse to work under the new conditions (assuming they are legal), then the State would generally consider that a "quit." In other words, it would be the same as an employee not showing up to work when scheduled, even if they don't agree with the new scheduling.
That said, the State will look further into the situation. If the change in conditions were so extreme that a reasonable person would have had no choice but to refuse/quit, then the State would approve UI benefits. Sometimes, when there is a legitimate child care issue and the claimant (you) can prove that you had no other options for child care, that will be the issue that the State will agree makes your "quit" for "good cause" attributable to the employer. Otherwise, the other facts that you have provided would not be considered good reason to refuse the change in your conditions of employment and therefore considered a quit without good cause.
Even if this was a termination under State law, they would still deny your UI benefits unless they found the child care issue compelling since you refused to work the hours given to you. So, the only difference the quit versus termination makes for UI is who carries the burden of proof at an appeal hearing. You would with a quit and the employer would with a termination.
Please feel free to ask follow up questions if you have them. If none, if you could take a moment to leave a positive rating in the ratings box above, I will receive credit for assisting you today. Thank you