Why can't we require in the definition of a trivial fibration that the map has the right lifting property with respect to each inclusion of a closed subset into a (hereditary) normal space, or perhaps with respect to each $C$-embedded subset ?

What are natural examples of trivial Hurewicz between "nice" spaces which are not fibrations in this stronger sense ?

I suppose it might be harder to give examples of trivial fibrations in this stronger sense. But I am mostly interested to see some examples and counterexamples:

i. a closed inclusion $i:A\subset B$ into a (better if hereditary) normal space $B$ and a trivial Hurewicz fibration $f$ of "nice" spaces such that $i$ does not lift with respect to $f$

ii. a trivial Hurewicz fibration $f$ which does lift with respect to all closed inclusions into (hereditary) normal spaces

The only result I know is the obvious remark that the map contracting a retract of $\mathbb{R}^n$ to a point, is a trivial fibration in this sense. I suppose these questions should have been discussed somewhere but I was not able to find anything that does not assume that it is a closed inclusion into a paracompact space of finite Lebesgue dimension, or the fibration is a map to a single point.