Correct migration rate expression in infinite island model
Definition of $m$

There is no correct or wrong expression. It all depends what exactly you are interested in and what is the definition of m in a specific paper.

Def. 1

Some models assume that $m$ is the migration rate from a given population to another given population.

Def. 2

Some models assume that $m$ is the probability of migrating to any other population. In a two population case, this definition and the above are the same. In other case, the probability of migration to a given population is $\frac{m}{d-1}$

Def. 3

Some models assume a migration pool. With probability $m$ an individual is a migrant, goes to the migrant-pool and is then redistributed among all other populations (only works for island models). In such model, a migrant could be an individual that actually come back to the same population. So, the probability of really migrating is $\frac{m (d-1)}{d}$ and assuming equal migration rate among all populations, the probability to migrate to a given population is $\frac{m}{d-1}$.

Fst

If you're dealing with $F_{ST}$ in a finite island model, then you will have some $\frac{m d}{d-1}$ terms. I talk about it in this post but it is not complete, so you might want to read Slatkin (1991) and Charlesworth (1998)