You are misinterpreting the meaning of foreign owned.
A branch is foreign owned because it is incorporated in Singapore under the articles of incorporation of the foreign company, and it does not issue stock in Singapore.
Similarly, a rep office is foreign owned because it too, is a legal entity attached to the foreign company. It doesn't issue stock, either.
But, if you have formed a Singapore Pte Ltd and the foreign company holds all the stock (which is how virtually every foreign company with offices in Singapore arrange things), then you have foreign shareholders but the company is a Singapore registered company, it issues shares in Singapore, it is guided by its Singapore articles of incorporation, and therefore, it is a local company.
Foreign shareholders do not make a Singapore registered pte ltd "foreign".
Edited to add: If indeed your company is a Singapore registered pte ltd, and you are the director, then only you need sign the forms. In fact, the foreign holding company is nothing more than a shareholder, and under Singapore law, directors hold all the responsibility with only very few functions (like director remuneration and/or replacement) reserved for the shareholders.