Difference in requirements for functions to be linearly independent in $C[a,b]$ vs $C^{n-1}[a,b]$

If functions $f_1,\cdots f_n$ Are linearly independent in $C^{n-1}[a,b]$then they also are in $C[a,b]$.

But why is there a difference in (the definition of) linear independence for $C[a,b]$ vs $C^{n-1}[a,b]$?

The functions are the same, and the coefficients multiplying the functions for the linear combination are scalars, so where does the universal set (e.g. $C[a,b]$) come into play?

There is no difference. More generally, if $V$ is a subspace of a vector space $W$ and if $S\subset V$, then $S$ is linearly independent in $W$ if and only if it is linearly independent in $V$. This follows from the definition of linear independence.