You do need to report the sale of your home. The section of the IRS Publication you included in your post... "your gain that is eligible for exclusion from your income is not to be reported on your tax return..." only applies if there is a period of non-qualified use. Non-qualified use is that period of time during which the home you sold was not your primary residence.
It is also important to report your home sale on your tax return because you can exclude any capital gain--up to the limit--only once every two years. Thus, your tax return can serve as your evidence as to the last time you excluded a capital gain in connection with the sale of a home.
@ConnorJ88
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