I've got a couple brands that I've used, but don't know how to assess the safety well; I'll let others speak to that, or perhaps to which rating organizations they'd trust.

[Edit: I strongly agree with your point about how you use the heater.] safety is not just about the device, but also how it's used. I try not to leave a high-current appliance unattended: if I must leave an oil-filled heater on and unattended, I set it to low or maybe medium power, greatly reducing the current; I reduce the amperage on my level-2 car charger, unless I want to charge promptly during a short stop at home. And, don't overlook all those safety warnings that we don't read in the instruction booklet... Occasionally check the cord for wear, and make sure it doesn't get damaged. After your post, I may start feeling the cord after the first 5-10 minutes of use, to see if it's heating excessively or unevenly.

And, beware the phrase "all other things being equal", even when it is unspoken. The oil-filled heaters are said to be safer, but that's if you use them in the same way ... they take time to warm up, so it's much more tempting to turn them on well before you're going to occupy the room. Would it be safer to start an old-style electric room heater only when you're actually in the room? I don't know, but that's a different question, any may have a different answer. [Edit: So, maybe something with a remote control, so you can shut it easily, each time you step out?]

Also, I try not to put the heater between myself and the door. Just in case something gets out of hand quickly.