How do people who believe in "no self" live?

If your use a definition of a word that is not the most common, then it’s you who should point that out. The “core meaning” below is the closest match to what you are described just now, but it’s not saying soul is something that is physical or could be mapped out in the human body. That’s what you want. It’s what you say bothers you if you can’t do that mapping.

But, you use language that fits better with “component of personality” for your arguments that soul does exist. And it does exist as that concept, that’s how we describe what we feel. Why you want more than that, you don’t say.

A soul is generally defined as the immaterial essence, spirit, or immortal aspect of a human being, distinct from the physical body . It encompasses consciousness, personality, emotions, and intellect, often considered the seat of human consciousness and moral character.

Key Aspects and Definitions:

Immaterial Essence: Often viewed as the “inner” person that continues to exist after the death of the body.

Component of Personality: Includes the mind, will, and emotions, driving human behavior and identity.

Religious/Philosophical View: Represents the spiritual principle that unites the body and spirit, frequently considered to be created by God.

Core Meaning: It can represent the essential part or core of anything (e.g., “the soul of honor”).

It’s not component of personality, rather it’s unified person. A self of sorts, something that you could say is some “one”. The problem is a lot of philosophy I’ve read lately argues against the notion of a person “existing”.

The second argument from metaphysics to ethics discussed by Lele draws on Śāntideva’s arguments that are intended to refute the real existence of the body. These are not offered merely to satisfy some form of pure intellectual curiosity; they are intended to have important motivational effects. Many people are proud of their bodies, and almost all adults have felt sexual attraction towards the bodies of some other people. Both the pride and the attraction, Śāntideva holds, can operate only when we ignore the insubstantial nature and disgusting character of our own and others’ bodies (see, e.g., BCA VIII.30–70; ŚS 231). Moreover, ordinary people understandably feel intensely protective of their own bodies, and this attitude often leads to morally questionable behavior. According to Śāntideva, if we clearly understand what the body is, we can transform our emotional relationship with it. This transformation involves coming to see the body as “like a machine, a mechanical construct of interconnected bones and muscles” (ŚS 231, SS-G: 225). Though it has no intrinsic value at all, the body is instrumentally useful as a means to support spiritual practice and to assist other sentient beings; as such, it makes sense to protect the body as long as it can play a useful role. And when it can no longer do so, it can still become food for carnivorous animals.

Considerable scholarly attention has recently focused on the third of Śāntideva’s arguments, from the absence of any self to the rationality of altruism. This argument was already explored in an insightful way by Luis Gómez in 1973, and became a major topic of discussion largely as a result of the criticisms directed at it by Paul Williams (see Williams 1998, Siderits 2000, and Williams 2000). The argument continues to attract both critics and defenders (see especially ch. 4, 6, and 8 of Cowherds 2016).

Though these two offer a more balanced perspective:

Reading the deconstruction of the body | Love of All Wisdom.