If a hydrogen atom loses its electron then it is then usually referred to as a proton, although it might be something else depending on the hydrogen isotope involved.

(1) Nomenclature for Hydrogen Proposed. Chem. Eng. News Archive 1986, 64 (29), 27. https://doi.org/10.1021/cen-v064n029.p027.

If a hydrogen atom loses its electron then it is then usually referred to as a proton, although it might be something else depending on the hydrogen isotope involved.

The following summarizes the recommended nomenclature, adapted from a webpage on the subject (see also the authorative references [2-3]):

(1) Nomenclature for Hydrogen Proposed. Chem. Eng. News Archive 1986, 64 (29), 27. https://doi.org/10.1021/cen-v064n029.p027.

(2) IUPAC-IUBMB Joint Commission on Biochemical Nomenclature (JCBN) and Nomenclature Committee of IUBMB (NC-IUBMB). Journal of Molecular Biology 1998, 275 (3), 527–537. https://doi.org/10.1006/jmbi.1997.1433.

(3) Bunnett, J. F.; Jones, R. A. Y. Names for Hydrogen Atoms, Ions, and Groups, and for Reactions Involving Them (Recommendations 1988). Pure and Applied Chemistry 1988, 60 (7), 1115–1116. https://doi.org/10.1351/pac198860071115.

You make some false assumptions, for instance that in water hydrogen atoms give away their electrons and oxygen ends up with 10 electrons. In molecules electrons are usually "shared". They don't "belong" to anyone. It is a meritocracy of sorts. Electrons may display a preference for some parts of a molecule, such as the regions between nuclei, so you could say they "hang out" there more.

If a hydrogen atom loses its electron then it is then usually referred to as a proton, although it might be something else depending on the hydrogen isotope involved.

IUPAC gave some attention to the issue of hydrogen nomenclature and provided guidelines:

Under IUPAC's nomenclature proposal, the prefixes hydro- and hydri- would be used to refer to the mixture of hydrogen isotopes that occurs in nature. Proto- and proti- would be reserved for the $\ce{^1H}$ isotope, either alone or as a group. Thus, the word proton would mean the $\ce{^1H+}$ ion exclusively and not $\ce{H+}$ in natural abundance. The $\ce{H+}$ cation would be called the hydron. Similarly, the $\ce{^2H}$ isotope would always be referred to using the prefixes deuteri- or deutero- (as in, for example, potassium deuteride, $\ce{K^2H}$). The $\ce{^3H}$ isotope would always have the prefix triti- or tritio-.

I remember a lecture from which my then supervisor left enraged because the speaker had repeatedly referred to hydrogen atoms as protons (I believe it was in the context of NMR, although the talk was about formulation). Despite the difference, the terms are used to mean roughly the same thing in some contexts, specifically in NMR, where it is the behavior of the spin associated with the $\ce{^1H}$ nucleus - the proton - that is most commonly interrogated, although deuterium NMR is very much possible.

(1) Nomenclature for Hydrogen Proposed. Chem. Eng. News Archive 1986, 64 (29), 27. https://doi.org/10.1021/cen-v064n029.p027.

You make some false assumptions, for instance that in water hydrogen atoms give away their electrons and oxygen ends up with 10 electrons. In molecules electrons are usually "shared". They don't "belong" to anyone. It is a meritocracy of sorts. Electrons may display a preference for some parts of a molecule, such as the regions between nuclei, so you could say they "hang out" there more.

If a hydrogen atom loses its electron then it is then usually referred to as a proton, although it might be something else depending on the hydrogen isotope involved.

IUPAC gave some attention to the issue of hydrogen nomenclature and provided guidelines:

Under IUPAC's nomenclature proposal, the prefixes hydro- and hydri- would be used to refer to the mixture of hydrogen isotopes that occurs in nature. Proto- and proti- would be reserved for the $\ce{^1H}$ isotope, either alone or as a group. Thus, the word proton would mean the $\ce{^1H+}$ ion exclusively and not $\ce{H+}$ in natural abundance. The $\ce{H+}$ cation would be called the hydron. Similarly, the $\ce{^2H}$ isotope would always be referred to using the prefixes deuteri- or deutero- (as in, for example, potassium deuteride, $\ce{K^2H}$). The $\ce{^3H}$ isotope would always have the prefix triti- or tritio-.

I remember a lecture from which my then supervisor left enraged because the speaker had repeatedly referred to hydrogen atoms as protons (I believe it was in the context of NMR, although the talk was about formulation). Despite the difference, the terms are used to mean roughly the same thing in some contexts, specifically in NMR, where it is the behavior of the spin associated with the $\ce{^1H}$ nucleus - the proton - that is most commonly interrogated, although deuterium NMR is very much possible.

References

(1) Nomenclature for Hydrogen Proposed. Chem. Eng. News Archive 1986, 64 (29), 27. https://doi.org/10.1021/cen-v064n029.p027.