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What is the difference between stochastic discount factor and stochastic discount factor process and how are they both related?

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You should consider the following different, but related, concepts:

-Pricing kernel: It is a stochastic process $\{M_t\}_{t=1}^\infty$ such that the pricing process is a martingale, i.e., $M_t S_t =E_t[M_{t+1}S_{t+1}] \quad \forall t$

-Stochastic discount factor: you can divide both sides of the previous equation by $M_t$ to obtain that $P_t=E_t\left[\frac{M_{t+1}}{M_t}S_{t+1}\right]=E[SDF_{t+1}\times Payoff_{t+1}]$ . The SDF tells you about the ratio between marginal utility of wealth at time t+1 and at time t, i.e., $SDF_{t+1}=\left(\frac{\partial U( W)}{\partial W_{t+1}}\right)/\left(\frac{\partial U( W)}{\partial W_{t}}\right)$

-Stochastic discount factor process: it is the stochastic process of $\left\{SDF_t\right\}_{t=1}^\infty$

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