= Solution
Use the following normalizations for <Hecke operators on marked lattices> and the <diamond operator>:
$$
\boxed{(T_pF)(L,t)=\frac1p\sum_{\substack{L'\supset L,\ [L':L]=p\\\operatorname{ord}(t\bmod L')=N}}F(L',t\bmod L'),\qquad
(\langle d\rangle F)(L,t)=F(L,dt).}
$$
Multiplication by a unit $d$ preserves the order of the marked point. There are $p+1$ prime-index overlattices; when $p\nmid N$ they all preserve that order. When $p\mid N$, precisely the overlattice killing the order-$p$ <subgroup> generated by $(N/p)t$ is excluded, leaving $p$ terms. Thus the definition is meaningful at bad primes too.
Both operations preserve homogeneity, since scalar multiplication bijects the indexing lattices and multiplies every summand by the same $u^{-k}$. Changing the marked basis merely permutes the overlattices, giving the required $\Gamma_1(N)$ transformation. Locally each term is a <modular form> evaluated after a rational fractional-linear substitution, multiplied by its appropriate automorphy factor; this preserves holomorphy on the half-plane.
For <modular cusp> holomorphy, factor any such rational substitution at a rational <modular cusp> into an integral modular substitution followed by an upper triangular map $z\mapsto Az+B$ with $A>0$. An existing holomorphic <modular cusp> expansion stays bounded under this map. The finite sum has the positive period supplied by its new level, so boundedness makes its singularity removable in the new <modular cusp> parameter. This is <cusp holomorphy under rational slash operators>. It proves that the operators preserve the functions arising from $M_k(\Gamma_1(N))$.
For clarity, the factor $1/p$ is paired with the homogeneity convention $F(uL)=u^{-k}F(L)$; it gives the Fourier normalization requested in part (c). The diamond action on <modular forms> equals $f|_k\gamma$ for a lift $\gamma\in\Gamma_0(N)$ with lower-right entry congruent to $d$, since the transformed marked point is $d/N$ modulo the original lattice.
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