If Heisenberg uncertainty principle involves the standard deviation of quantities then why do we use it in a different way as here?

The first formula is more precise. For a gaussian wave package the standard deviations in x and p are related by this expression.

A wave function that is uniform over a spherical volume and zero outside it has high momentum components due to its sharp edges. This causes he product of x and p standard deviations to be larger than $hbar/2 = h/4pi$.

The idea is that if a particle can be anywhere in the sphere of radius $Delta R$, then $sigma_xsimDelta R$ (you can try to calculate the standard deviation of the position of a particle that can be anywhere on a sphere of radius $Delta R$ and it will be proportional to $Delta R$). It is a bit like the Fermi Problem (https://en.wikipedia.org/wiki/Fermi_problem), in which you are only interested in estimating the order-of-magnitude of something.

This kind of "estimates" are not rigorous but common. However, this is usually enough. Notice that, independently of the rigor of the inequality, the physical interpretation will remain the same: if the radius of this sphere to which the particle is confined decreases, the uncertainty of the momentum increases (i.e. it is very hard to confine particles to very small spaces).

This answer is a comment really.

The standard deviation has a strict statistical meaning

The root-mean-square deviation of x from its average is called the standard deviation. For a set of discrete measurements, the standard deviation takes the form

for continuous:

....

Determining the average or mean in the above expression involves the distribution function for the variable.

The distribution function is also statistically defined, for example:

, Poisson

That is why the $σ$ symbol is usually confined to standard deviation.

The Heisenberg uncertainty(HUP) is usually given as

with the $Δ$ symbol instead of the $σ$ to keep clear that the distribution function is not one of the statistical ones, but given by the quantum mechanical solution of the equations and boundary conditions of the problem, $Ψ^*Ψ$, a probability distribution, but not a statistical one.