There are three basic families of restricted compositions (ordered partitions) that are enumerated by the Fibonacci numbers (with offsets):
a) compositions with parts from the set {1,2} (e.g., 2+2 = 2+1+1 = 1+2+1 = 1+1+2 = 1+1+1+1)
b) compositions that do not have 1 as a part (e.g., 6 = 4+2 = 3+3 = 2+4 = 2+2+2)
c) compositions that only have odd parts (e.g., 5 = 3+1+1 = 1+3+1 = 1+1+3 = 1+1+1+1+1)
The connection between (a) & the Fibonacci numbers traces back to the analysis of Vedic poetry in the first millennium C.E., at least (Singh, Hist. Math. 12, 1985).
Cayley made the connection to (b) in 1876 (Messenger of Mathematics).
$\bullet$ Who first established the connection with (c), odd-part compositions? It was known by 1968 (Hoggatt & Lind, J. Comb. Th.), but I suspect it was done before that. Thanks for any assistance, especially with citations.
By the way, it is a nice exercise to give combinatorial proofs of why each family is counted by the Fibonacci numbers, and establish direct connections between each pair of families.
PS: Apologies for cross-posting from MathOverflow, want to see if the audience here has more knowledge of such things.