What is the official, generally accepted definition of "lightweight crypto"?
There is no official definition, of course. However, at least in my experience, the word is used consistently.

The word lightweight typically refers to something that is significantly less expensive to use than other stuff, while at the same time achieving the same desired effect, but this lightweight-ness has some cost. (If there was no cost to lightweight-ness, it would not be lightweight, but just "better".) This means that the lightweight stuff may be desirable in some situations, but not in others.

Typically, a lightweight block cipher will be amenable to implementations that are small or use little power or ..., compared to the usual block ciphers. This happy state is achieved at some cost, perhaps by reducing the security margin or having more rounds, etc.

Note that people may talk about lightweight public key cryptography, but it is lightweight only when compared to ordinary public key cryptography, not compared to other kinds of cryptography such as block ciphers.

People also talk about lightweight implementations of cryptographic primitives. Again, lightweight refers to some reduced cost, typically achieved by increasing some other cost. People may be interested in an AES implementation with extremely small gate count, for instance, even if it would be slower or use more power than other implementations.

The word lightweight is of course not limited to cryptography, but is used everywhere, with somewhat differing meanings. Consult your dictionary. In technical contexts, it usually has positive connotations in my experience.