I mean it depends doesnt it, really? My bf is is SAHM to two teens. She is a lunchtime supervisor just for fun. She has health issues and SEN so most FT jobs would be impossible or unbearable (painful) for her. Her life is pretty easy - she has a cleaner to do the heavy housework. Her DH provides for her. She spends a lot of her time socialising or gently exercising to help her health conditions. She is very happy, and has bags and bags of free time and enough money to enjoy her leisure time without being excessive.
On the other hand I know SAHMs who have at least one child, a DH who works all hours and does nothing to help run the household, and caring responsibilities for elderly relatives, are scraping by financially and exhausted all the time.
The idea that you can compare these two lifestyles is silly.
I am a FT mum with a DH who also works FT. I have a cleaner to help once a week as we find that easier albeit expensive. Do I think I work hard? YES!!! I work exceptionally hard even though I have avoided promotion so I could prioritise work.I want to earn a pension for myself. I want my kids to enjoy extra curriculars but still have plenty of free time with me. I want to have it all.
Would I say I work harder than the average SAHM? No because I simply don't believe in an average SAHM.
Would I say the language needs changing? No, because I understand the conventional short-hand "I don't work" means "I don't have an independent source of income generated by employment or self-employment". That's just a ridiculous mouthful.
When I was a SAHM (for three years) I rarely said "I don't work". Typically I said, gleefully to my working parent friends "I've got a few years off work to spend with my elderly mum and my kids, and I'm broke but I'm loving it!". It was definitely, definitely a much easier life than being a working mum, even with a needy baby/toddler and a dying mother to care for.
The phrase "my DP is the sole earner" is factually accurate. I don't see how you can argue with that. Simply because the SAHP is facilitating the arrangement whereby one partner works to generate the household income, doesnt mean the non-earning partner can claim to be earning.