Non-contact tester shows voltage in house circuit while three-lamp tester shows none
Non contact testers work by identifying an electrical field. The wiring normally is run in close proximity and the live wires induce a voltage that can be detected by the non contact tester but there is no potential or current available so the lighted tester will not light. This is normal and quite common I have more expensive Fluke testers and very cheap ones and they all work the same. You can duplicate this with 2 extension cords plug 1 in and put another one not plugged in next to the first. The larger the load on the plugged in cord the larger the field will be on the cord that is not plugged in but many times no load is needed to induce the field in just a few feet. (this is how transformers work but they have many wraps of wire to develop the potential).
Have you checked your circuit breakers? if the handle is partially tripped it needs to be moved to the off position then back to the on position to reset it. it is also possible that the circuit is protected by A GFCI device that has tripped other than a failed electrical connection.