D1: Let $(a_n)$ be a sequence. We say that $(a_n)$ converges to $L$ if for every $\epsilon>0$ there exists a natural number $N$ such that for every $n>N$ we have $|a_n-L|<\epsilon$.
D2: Let $(a_n)$ be a sequence. We say that $(a_n)$ converges to $L$ if for every $\epsilon>0$ there exists a natural number $N$ such that for every $n\geq N$ we have $|a_n-L|< \epsilon$.
D3: Let $(a_n)$ be a sequence. We say that $(a_n)$ converges to $L$ if for every $\epsilon>0$ there exists a real number $N$ such that for every $n>N$ we have $|a_n-L|<\epsilon$.
D4: Let $(a_n)$ be a sequence. We say that $(a_n)$ converges to $L$ if for every $\epsilon>0$ there exists a real number $N$ such that for every $n\geq N$ we have $|a_n-L|<\epsilon$.
The definitions are equivalent. Why would one use one definition over the other, is it just a matter of convention? If this question it too general please provide guidance on how I can improve it, thanks.
When we are dealing with indices of a sequence it is peculiar to compare them to real numbers, so 3 and 4 are strange. The difference between 1 and 2 is just $1$ in the minimum $N$ you use. As we only care about whether there is an $N$, not what it is, it doesn't matter which you use.