Under my agreement with my current employer, they retain ownership of everything that I create, both at work and side projects (the legal wording is very lose on this, and I guess it's like that for a reason).
I can't completely quit my job to try my hand at a startup idea, so I must get the MVP done and off the ground on the weekends.
What can I do to ensure I can do this legally? Could I just remain anonymous (not list myself as a founder of the app) and just quit my job if the app becomes profitable? And continue to remain anonymous in order to avoid any legal conflict.
Anonymous is not the way to go. If the legal agreement is valid, you could still get in trouble once you side project starts succeeding.
You really need to check with a local lawyer (initial consultations are usually free, so get an appointment with several). Often, what is written in the agreement is actually broader than what the local laws allow. It's typical for an employer to try to overreach.
Unfortunately, sometimes, your country/state will actually enforce very broad agreements. Again, working with a lawyer, read the fine print. Maybe you need to notify your employer ahead of time. If you work for a very large company, sending an e-mail to some HR person far away might protect you. Again, talk to a lawyer.
Alain has some great points, but I want to add one more thing. If you can't get around your contract, there's another option.
You may not be allowed to work on another project, but you're not prevented from cutting back your expenses and saving up as much money as you can over the next year or two. Basically build a 6 month, 1 year, or multi-year runway in savings to give yourself time to work on your startup after you quit. Then quit, and dedicate all of your efforts to building your startup after.
Many people find that they don't have the time nor the focus to do what's needed to get a startup going in their spare time anyway. That's why you'll see a lot of people saying things like, "You're not serious about your startup until you quit your day job." Or, "You can't get funding unless you're working on this full time." While I don't agree with those ideas, there's validity to the point that after a long day of work, you've already expended your best mental strength.
So yeah, another option is to just do everything you can to save cash for the next X months/years, so that you can quit and have Y months/years completely free to work on your startup.