(Updated to reflect John Shareshian's excellent answer and suggestions.)
Yes, there are infinitely many examples.
First off, some negative results: If $[G:H] ≤ 31$, then $G/Core(G,H)$ acting on H is one of your three examples. If H is contained in a core-free maximal subgroup M with $[G:M] ≤ 50$ (subject to a few caveats), then in fact G is your first example. These "$M$" correspond to John Shareshian's type (2) examples.
However, I think your search of small groups must have had an error:
The group $G =$
SmallGroup(648,725) has Sylow $2$-subgroup $H$ that is core-free, and the interval $G/H$ is the bounded version of a $4$-element anti-chain.
This corresponds to the next smallest $H$ (after $H≅2$) in John Shareshian's type (1) examples.
As a general comment, no interval is so rare that it only occurs finitely many times:
Given any interval $[G/H]$ there is an anti-isomorphic interval in a wreath product of the permutation group $(G,H)$ with a non-abelian simple group.
Applying the construction an even number of times produces an infinite sequence of examples with a given (core-free) interval. In particular, there is an example with $|G|=279936000000$ and $|H|=360$, with $H$ core-free and the interval $G/H$ the bounded version of a $4$-element anti-chain, $M_4$. This takes the "seed" $(G,H)$ to be the regular representation of the non-abelian group of order $6$, and the non-abelian simple group of order $60$.
This is very similar to F. Ladisch's answer to your previous question.
Here's what I've found in the literature (both of which are referenced in Roland Schmidt's book; I did not find much else):
Kurzweil (1985) puts your second and third example into context ($G/N=H$ acting on an isotypic semisimple module $N$, so that the interval $[G/H]$ is a projective space). Your first example is just "small", I think. More importantly, it gives the method of replicating examples using wreath products as the second example on page $148$.
Heineken (1987) pins down the structure of solvable $G$ with second maximal $H$ such that the interval $[G/H]$ is not a (bounded) antichain, $M_n$. He has some results that say $n−1$ is usually a prime power.
John Shareshian points out that Baddeley–Lucchini (1997) is dedicated to the question of which $M_n$ can occur as intervals in the subgroup lattice of a finite group. In particular, this shows that the $n−1$ being a prime power result is definitely restricted to solvable groups: Feit showed both $n=7$ and $n=11$ occur. The paper is definitely focused on the non-solvable case.
The Math Review is also really good.
"Endliche Gruppen mit vielen Untergruppen."
J. Reine Angew. Math. 356 (1985), 140–160.
"A remark on subgroup lattices of finite soluble groups."
Rendiconti del Seminario Matematico della Università di Padova, 77 (1987), 135-147
Baddeley, Robert; Lucchini, Andrea
"On representing finite lattices as intervals in subgroup lattices of finite groups."
J. Algebra 196 (1997), no. 1, 1–100.