(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.

Kurzweil, Hans.
"Endliche Gruppen mit vielen Untergruppen."
J. Reine Angew. Math. 356 (1985), 140–160.
MR779379
DOI:10.1515/crll.1985.356.140
GDZ:GDZPPN002202190

Heineken, H.
"A remark on subgroup lattices of finite soluble groups."
Rendiconti del Seminario Matematico della Università di Padova, 77 (1987), 135-147
MR904616
NUMDAM:RSMUP_1987_*77*_135_0

Baddeley, Robert; Lucchini, Andrea
"On representing finite lattices as intervals in subgroup lattices of finite groups."
J. Algebra 196 (1997), no. 1, 1–100.
MR1474164
DOI:10.1006/jabr.1997.7069