Confusion over equations for population's heterozygosity vs. heterozygous individuals
One is a recurrence equation and the other is a general solution. These are basic concepts of mathematical modelling. 

Reccurence equation

A recurrent equation describes the state of a system (here, heterozygosity $H$) at the next time step given the state in the previous time step. The recurrence relation is $H' = (1-1/2N)\times H$. Given the state $H$, at a given generation, one can know the state $H'$ at the following generation (given the population size $N$).

General solution

A general solution describes the state of a system at any time step given an initial state. Not all recurrence relation have a general solution. Here there is a general solution and it is $H_t = H_0 \times (1-\frac{1}{2N})^t$. Given the initial state $H_0$, one can know the state $H_t$ after $t$ generations (given the population size $N$).

Two equations for the same model

The two equations both refer to the same model and therefore the same statistic of expected heterozygosity which, as you said, is $H = 2 p (1-p)$ (for a bi-allelic locus) by definition.

Understanding the basics of mathematical modeling in ecology and evolution

Have a look at A Biologist's Guide to Mathematical Modeling in Ecology and Evolution by S. Otto for an introduction into mathematical modeling in the field of interest.