While it's often hard to define exactly what is the very first of some particular technology, I believe many people claim the PDP-1 was a minicomputer and thus would trump the LINC and PDP-5. Though probably the actual term "minicomputer" was not yet invented when the PDP-1 was introduced. It was of course physically larger than the 8 or 11 due to the available technology, but architecturally of a similar size. It was also quite inexpensive compared to other computers in the late 50s.
I have seen 60s literature refer to "medium size computers". This seems mostly forgotten today, with only mainframe and mini categories being in use to describe machines of that era. But it seems a useful distinction between a mini which is just a cabinet or a few boards, and a hulking room-sized mainframe with a plethora of subordinate I/O processors.