You are 'authorized to stay', which also means you have 'lawful presence'. However, you don't fit into either 'immigrant status' or 'nonimmigrant status', so you're not eligible to 'change status' into one of the many visas, nor 'adjust status' into LPR.

So remember, the word 'status' has a specific meaning for immigration purposes. They're correct when they say you don't have legal 'status', but you are not illegal nor undocumented anymore. They don't give EAD and Advance Parole to illegals.