The wife wants a head torch for running during the darker evenings with the dog so it needs to be bright enough to see by and she'd like a red light on the back too.
What do you recommend?

I use an Alpkit Qark which is 300 lumen, rechargeable and has the red safety light on the back. Alpkit have a new range for this winter so it's reduced as well.

https://www.alpkit.com/products/qark-1

I bought one of the fancy petzl reactiks for night running last year. Great bit of kit. Rechargeable and the functions work really well. Automatically changes light spread & power according to your view i.e. flood close up and spot looking into distance. Looks bulky and only a single strap head-band but rock solid, comfy etc. Never wanted for more power in the pitch dark woods.

Only thing I've not really needed or used is the Bluetooth ability to connect to phone.

Price is a bit steep but if I lost mine I'd not hesitate to buy another to replace.

Maybe something simpler and cheaper would do the job but this is only one I've used seriously and it's been great.

Edit - no rear red light though. Do Alpkit still do the little button lights?

The reduced AK Quark dies the rear red light but, having briefly tried one on, I preferred the weight distribution of the Petzl. Mucho cheaper tho.

Used an old Alpkit gamma to run in dark last year when Marathon training. Out of all my head torches it worked best for running because of the middle strap.

I've a Petzl Reactik as well, it's a really good headtorch. You can download the Petzl software which lets you play around with custom settings for the different modes. I use mine for running and xc skiing mainly, the only issue I've found is that on cold nights when my breath fogs, it can confuse the sensor, making it switch to "low" mode. Then I just switch it to constant and it's fine. It has a good strap and a nice wide base so it is very stable, doesn't bounce around at all.