Question regarding social welfare in the presence of externalities I am currently learning high school level economics and I do not understand how indirect taxation may increase net social welfare in the presence of negative production externality.

This is because, from analysis of welfare in the market diagram with linear demand and supply curves, I found that the social welfare gain due to a tax correcting a negative production externality is equivalent to the welfare loss to consumers and producers from the taxation itself (resulting in no net change in total welfare).

Theoretically, this can also be explained by the fact that taxation reduces the net satisfaction of consumers, firms, and government, but increases net satisfaction in the 3rd party.

My current understanding is that correcting externalities do not cause net welfare to change, but it allows the socially optimum level of resources to be allocated to production, answering the "what to produce question" and allowing the economy to benefit in the long run, but I'm not at all sure.
Your understanding is close, but not completely correct. Look at the picture below. Blue is the demand curve, Orange the marginal private costs (which in perfect competition is equal to supply). Should there be no taxation or externality the equilibrium price would be P1 and the equilibrium quantity Q1.

If there is a production externality the social costs of production are higher, illustrated by the marginal social costs in grey. The total costs to society (or the 3rd party) of the externality is the area A+B+C.

If the government introduces a tax equal to the external costs of production (the tax rate per unit would be the vertical distance between the orange and grey curve) we would achieve a new equilibrium with a new higher equilibrium price P2 and a lower equilibrium quantity Q2.

The total tax burden for the company is area A. This burden was formerly carried by society and is thus not a net gain, but a transfer. Area B is also a transfer. The upper part above the black line comes from the consumers, the bottom part comes from the producers. There is more re-allocation of surplus going on, on the left hand of the dashed line because prices are higher and quantities are lower, but I haven't illustrated those to keep things simple.

Area C though is a gain. That part of the costs are no transfer from anyone and were borne by society before the externality got corrected (or again the 3rd party)

In conclusion introducing a socially optimal tax level introduces a lot of transfers, but also a net gain in welfare.