The situation you're describing is pretty risky and not well covered legally. If your contract says "on-site" but you're working remotely, there's a gap between what the paper says and reality, and that leaves you vulnerable. If something goes wrong later - a workplace accident at your house, a firing, a conflict over hours or responsibilities - the company can use that old contract in their favor and you're left without documentation backing up how you're actually working.
What you should do is ask in writing (email, message, whatever but something on the record) to update your contract to reflect remote work. It's not you being picky, it's mutual protection. Specify the terms: whether it's 100% remote or hybrid, what the hours are, where you're keeping company documents, how confidentiality is handled in your home, all of that. If the company refuses or lets it slide, at least you've got that email where you asked to formalize things - that protects you.
Some people say they've been through this and it ended badly, and it's because they let time pass without a document. Don't wait, don't let them "get used to" you working remotely without updating anything. The more time that goes by like this, the harder it gets to argue later what the actual arrangement was. Move on it now while everything's running smoothly, it's way easier than negotiating when there's already a problem.