How right is defining elements with the number of protons? Atomic number is the number of protons a certain atom has. It's the defining attribute of a certain element. But, How right is that?

From the viewpoint of chemistry, the definition is right. Because, for a certain chemical reaction, isotopes of a certain element will react in the same way.

For example, $\ce{H2 + 1/2O2 -&gt; H2O}$
$\ce{ ^2H2 + 1/2O2-&gt; ^2H2O}$

So, it's useful to define elements in terms of their number of protons.

But, other than reactivity in chemical reactions, many properties of an atom differ wildly due to difference of  number of neutrons and electrons eg. half life of radioactive decay, type of radioactive decay, magnetic moment, stability, nuclear quantum states etc. So, even having the same number of protons can't ensure much similarity in some aspects.

I think to a rational perspective, the similarity of reactivity of isotopes in chemical reactions should weigh more than the other nuclear difference between isotopes. So, in that logic, the definition is right.

But, difference in nuclear properties should do matter. So, how right is defining elements with atomic number i.e. number of protons?
Most of chemistry doesn't care a whit about all of the properties you named: they're in the realm of nuclear physics, as noted by user badjohn.  Even radioactive decay is something of a black sheep, included in physical chemistry as a convenient example of a first-order kinetic process. To my knowledge, it's not substantially used in conjunction with any other sub-discipline of chemistry, as it'd be more of an annoyance than anything ("Gah, the cobalt center of my complex keeps changing into nickel!").  It does of course have plenty of applications (medical imaging, nuclear power, etc.), but they're arguably in the realm of physics, with any attendant chemistry being a secondary consideration.

Magnetic moment, neutron count (isotopes), and nuclear quantum states do have some slight influence over certain spectroscopic properties (primarily NMR, as far as I know), but other than the somewhat special cases of the spin isomers of hydrogen and the kinetic isotope effect of deuterium vs. protium I'm not aware of any appreciable effect of these on chemical reactivity.

For the chemist, the identity of a nucleus is sufficiently defined (within an experimentally measurable perturbation, at least) in virtually all cases by its atomic number.