When should I down level myself on purpose at a new company?

It is fine to down-level yourself, and in some cases I'd actually encourage people to do this anyway for very senior roles in order to have less stress and more work/life balance.

However, if you're debating between L4 (mid-level) and L5 (senior), I'd push for the more senior level:

• At many companies, L5 (senior) is a terminal level, so you need to get promoted to that level within X months in order to remain at the company. So coming in at L5 may actually be less stress.
• There are many L5 engineers in Big Tech. So it's not like you'll have a huge burden of leading a 10 person team and charting a year-long roadmap. I know many L5s who have a really chill life without too much responsibility.
• If you're coming in at a large, well regarded company (e.g. Google), they'll usually give you ample time to ramp up. You can do it and perform well! (and Taro is here to help)

Mostly what Rahul said. If your level is L5 or below, it's better to avoid down-leveling. If you can negotiate up-leveling that's even better. If your level is above L5, down-level may help make your ramp-up easier.

The job responsibilities for L3 to L5 are pretty predictable. If you can do it you should be able to consistently do it forever. Down-leveling just means "we doubt you can do it consistently". It doesn't make a lot of sense. The only scenario in it that may make sense is you were promoted recently. They may doubt that you were promoted immaturely. But if you are confident you can do the job consistently you should push back.

The job responsibilities for L6 and beyond can be pretty different across companies. What you succeeded in doing in a previous company might not translate well in a new company. (From a hiring manager's perspective: It's always a high risk to hire an L6+ into the team.) Resetting back to a more predictable L5 (or a lower level that's above L5) makes sense. It gives you time to figure out whether the responsibilities are the ones you expected or had experienced. If not, it gives you time to learn the new responsibilities.