The English words "prayer" and "worship" just miss the idea

 The English words "prayer" and "worship" just miss the idea.  It's not unlike the mash about Chinese dragons and Western dragons: not the same beast at all.

Yes—that’s a very good analogy. The problem is not merely that English lacks a better translation; the imported word can bring along an entire conceptual framework that isn't there in the original.

“Prayer” in ordinary English strongly suggests a person addressing a deity—asking, thanking, confessing, listening for an answer. “Worship” suggests reverence or adoration directed toward a divine being. So when we use either word for Buddhist practices, we can accidentally smuggle the Christian (or at least theistic) model into Buddhism.

And your dragon comparison is apt. The Chinese lóng and the European dragon have acquired the same English label, but historically and culturally they're radically different creatures. One isn't simply a different species of the same thing; the word “dragon” encourages us to imagine a common category that wasn't really there.

I think the Buddhist case is even more interesting because some practices look like prayer or worship from the outside—bowing, chanting, incense, offerings, images of the Buddha—yet their meaning can be quite different. A Buddhist may be expressing reverence, recollecting the Buddha's qualities, cultivating particular mental states, making an aspiration, generating merit, or engaging in a ritual whose meaning depends on the particular Buddhist tradition.

So your childhood definition of prayer—a conversation in which an answer may be felt—is particularly useful here. That's a very specific conception of prayer, and it makes your instinctive reaction to calling Buddhist practice “prayer” quite understandable.

English speakers have a habit of saying, “Well, what else can we call it?” And sometimes the honest answer is: we can describe what they are doing, but there isn't a single English word that captures what it means to them.

We are obliged to use English words, but with a footnote.

Exactly. We can't escape English; we can only warn the reader that the English word is an approximation.

Something like: “Buddhist ‘prayer’—using the English term loosely; this does not necessarily imply petitioning a deity.”

That little footnote can make an enormous difference. It tells the reader, “I'm using this familiar word because you need one, but don't import everything you normally mean by it.”

In fact, that's how much cross-cultural vocabulary ought to work. Translation gives you the nearest handle; the footnote tells you where the handle doesn't quite fit.

And your dragon example is almost perfect for this: Chinese “dragon” (lóng): see footnote; not the European monster.