Lezyne are perfect. Don't get anything too bright.
That's the one I have (indeed I even have an unopened spare bought during one of those many deals). The only issue is it's easy to switch on accidentally when in bag and it can get rather hot!I can recommend the Moon Meteor X Pro and they're frequently on offer.
I'd agree with using two rear lights - and in terms of sheer VFM, it's hard to fault Poundlands 5-LED rear lights.
The lights come back on again when I press the switch again so I think the problem is here.You don't need a new light, just fix the problem with your existing light. If it is a battery light, it means that the batteries aren't tightly packed. Put a bit of foam rubber in to secure them. If it is a dynamo light, you have a loose contact somewhere.
The switch is just a springy metal strip that makes contact with a printed circuit. The first time it turns the light on, the next few times it goes through various flashing modes and then turns the light off.Loose contact. Disassemble switch and clean out.