So, it seems like the job runs under the user set as the
Yes, that is the account used (internally) to execute the code in the steps of the job, by default.
but I've read several questions here (1, 2) or in SO discussing "who is the user running the job", and none mentioned that.
The first two questions you've linked are unrelated. The first one is asking about finding who manually kicked off a SQL Job. The second one is asking who does the SQL Agent service account run as (externally, in Windows). The third question you linked is relevant but incomplete in information.
It is important to choose the proper
owner of your SQL Jobs. Generally that won't be an actual user, such as yourself, but should be generic dedicated Logins. I have a single, properly provisioned dedicated Windows account, which I use for the
owner of all of my SQL Jobs. That way, I don't have to worry about jobs failing if a specific person's account's permissions change, gets disabled (e.g. they leave the company), gets locked out (from password expiring, etc).