Is REINFORCE the same as 'vanilla policy gradient'? I don't know what people mean by 'vanilla policy gradient', but what comes to mind is REINFORCE, which is the simplest policy gradient algorithm I can think of. Is this an accurate statement?

By REINFORCE I mean this surrogate objective

$$ \frac{1}{m} \sum_i \sum_t log(\pi(a_t|s_t)) R_i $$

Where I indices over the $m$ episodes and $t$ over time steps, and $R_i$ is the total reward of the episode. It's also common to replace $R_i$ with something else, like a baselined version $R_i - b$ or use the future return, potentially also with a baseline $G_{it} - b$. However, I think even with these modifications to the multiplicative term, people would still call this 'vanilla policy gradient'. Is that correct?
By vanilla policy gradient, I think what is meant is normally any arbitrary policy gradient for the purposes of formalization(or whatever is trying to be communicated).

For example, let $J(\theta)$ be any policy objective function. Then our policy gradient would be $\begin{equation} \nabla_\theta J(\theta)  \end{equation}$ where the change in our policy is $\Delta \theta = \alpha \nabla_\theta J(\theta)  $ with $\alpha$ being our step size.

This will, of course, vary as notational convention and term definition can change between practitioners.