As for E-lokai Netzor and Hashem putting a soul within me, my own intent when saying these words is based on the Vilna Gaon's taxonomy of prayer:
Prayers that express an ideal to be repeated and internalized are what we call "tefillah" in Hebrew. Tefillos are consistently written in the plural, as our connection to the community is part of that ideal. Prayers written in the singular are therefore of a different sort, "tachanunim", expressions of what already exists in our hearts.
This is how the Gaon explains the line in Qaddish, "tisqabel tzelosehon uva'usehon -- accept the tefillos and requests (tachanunim) of all of the House of Israel..." We say this when closing the Amidah -- which is such a paragon of tefillah our Sages called it simply "Tefillah", E-lokai Netzor -- tachanunim, (note that it's written about "I" and "mine", not "we" and "our"), and Tachanun.
"Elokai, neshamah -- My G-d, the soul which you placed in me" is similarly tachanunim. Therefore, it's not a place to look for how we ought to see our self-definition, but how things feel to most of us first thing in the morning. The prayer reflects the fact that most people do in practice identify with our body most consistently, and only at times with their soul.
But to answer the philosophical question...
There is a machlokes, a dispute among the rabbis, as to how to view man. One side, found often among books of Mussar, views a person as a soul who inhabits a body, or perhaps controls it as a rider upon a donkey. As Elifaz describes humanity in the book of Iyov (4:10), “shochnei batei chomer – dwellers in homes of matter.” When Rav Yitzchak Isaac Scher (Cheshbon haNefesh, Slaboka Alumni ed., intro.) speaks of man’s physical side being an animal, we mean that literally, not merely like an animal. Since much of our yeitzer hara comes from our living in a mammalian body, R’ Scher recommends the very same strategies one uses for taming and being able to use the eyesight of a bird, the strength of an ox, the load bearing abilities of a donkey or the speed of a horse are applicable to gaining mastery over our bodies. Like any other animal, a person’s animal soul has no ability to plan toward a goal, it simply responds to whatever urge is most triggered in the moment. The animal soul must be saddled by the godly soul and guided. And Rabbi Sherr points out with the example of a trained elephant, “next to whom a person like his trainer seems little more than an ant”, to maximize its utility it must neither be overburdened or neglected, nor underused and let remind wild – and this is how we are to treat our body and our animal souls. Last and most importantly, neither an animal nor the animal within can be educated, but trained through habit and acclimation.
This notion is a key symbol in the Gra’s interpretation system — when one finds a chamor / donkey in a narrative, it is generally a symbol for the person’s chomer / physicality. Avraham at the Akeidah or the mashiach come in riding on a donkey as a way to indicate to us their mastery over their own physicality. In contrast, we speak of Bil’am’s donkey, but the Torah consistently calls it a different kind of animal; he does not harness a chamor, showing self-control over the animal’s urges of the moment, Bil’am rides an ason (Bamidbar 22:23,25,27,28,29,33).
In this viewpoint, a person is a rider of an animal, or to use a metaphor that may resonate better with our more modern lifestyles – the soul who is wearing a body.
Another stream of thought includes the body in the definition of person. Rather than a person’s more human side that rides his body as a master over an animal, in this model man is seen as a fusion of body and soul. For example when the gemara (Sanhedrin 91a) explains one purpose of the eventual resurrection of the dead by comparing a sinner to a blind man and a lame man who conspire to steal fruit from an orchard. They are caught and brought to court, but each of the accused claims innocence. The blind man says he must be innocent, for he was incapable of even finding the fruit, never mind stealing them. The lame man also claims innocence; after all, he had no way to reach it. Neither alone could commit the theft, so each of the accused points to the other as the critical element for the sin, the guilty party. The judge responds by putting one atop the other, recreating the unit that was capable of sin, and judges the pair. So too, the gemara explains, the soul could claim it couldn’t have sinned without the body giving it the opportunity for action, and the body could claim that the planning and execution of the sin are the fault of the soul. In order to judge us for our sins, Hashem will bodily resurrect the sinner to reconstruct the person as they were then.
As the Ramchal writes, “Man is different from any other creature. He is a combination of two completely diverse and dissimilar elements, namely, the body and soul.” (Derech Hashem 3:1:1)
The dispute is not necessarily about which is true, it could well be that both definitions of “person” are equally valid. The dispute is more prescriptive: When is it more productive to think of my physical aspect as an outsider, which would weaken the relative weight I would give the call of physical drives? And when am I better off not thinking of myself as purely soul, because then I’m not fully blaming myself for “stealing the fruit”?