The k from the footnote of section 1 is the required number of transactions to be referenced by new transactions (as is noted by you in your question).

The k from section 3 is just an index/placeholder in the context of number of tips overall (not the number of tips to be referenced), and one could have used any letter here. For example,
The limit of P[L(t) = m] as t → ∞ should exist and be positive for all m ≥1
conveys the same meaning.

The k from the footnote of section 1 is the required number of transactions to be referenced by new transactions (as is noted by you in your question).

The k from section 3 is just an index/placeholder in the context of number of tips overall (not the number of tips to be referenced), and one could have used any letter here. 

For example, The limit of P[L(t) = m] as t → ∞ should exist and be positive for all m ≥1 conveys the same meaning.