This is not an answer, but rather a personal experience. I'll post it as an answer since it's too long to be a comment.
If I would be an author, I wouldn't use the expression like "linear-approximation" since it's confusing.
I haven't seen an expression like "linear-approximation" or "logarithmic-approximation" for referring to approximation ratios. For example, the greedy algorithm for the minimum set cover problem is an $H(n)$-approximation algorithm, where $H(n)$ is the $n$-th Harmonic number, but I haven't seen it's called a logarithmic-approximation algorithm. Here, $n$ is the size of a universe, and this should be explained in the definition of the problem. Please note that $n$ is not the input size.
There's one exception. In the book "Complexity and Approximation" by Ausiello, Crescenzi, Gambosi, Kann, Marchetti-Spaccamela, and Protasi, I can find expressions like log-APX, poly-APX and exp-APX. For example, log-APX refers to the class of NP optimization problems that have an $O(\log n)$-approximation algorithm, where $n$ is the input size.
Some authors use "a linear approximation algorithm" to mean a linear-time approximation algorithm. I feel this is still confusing, and I would write linear-time approximation. People tend to omit "time", and for example when they write "a polynomial algorithm", they often mean a polynomial-time algorithm.