I have older house. The receptacle showed "open ground." I connected a wire from the green ground terminal to the nuetral terminal, and the meter read "correct." I left it that way because everything worked. Is it safe?

No it's not safe. Despite both the ground and neutral wire being terminated together in the fuse box/breaker panel, the neutral does carry current and a break in that wire between the box and the receptacle will cause all grounded parts of your appliance to be energized. Remove the wire from the ground lug and put a "No Ground" sticker on that plug. Better yet, consider using a GFCI plug there. It will detect differences between the current going in the hot wire and back the neutral wire and break the circuit if there's a difference (e.g, running through a knife blade into you child).