The question is in the title.

My current research subject is the homotopy theory of $2$-categories. It may be slightly unreasonable to expect my PhD thesis to be read or even looked at by people outside the field of higher categories, but for some reasons I would like this text to be enough self-contained so as to allow someone without any prior knowledge of $2$-category theory to understand at least the basic definitions. "Someone without any prior knowledge of $2$-category theory" does not refer to someone not knowing anything about category theory. However, I do not want to assume the reader to know anything more advanced than the very basic definitions of category theory. I therefore would like to be able to point out the reader to a text which does not require more advances prerequisites than elementary category theory. Otherwise I shall write the full definitions myself if no such text is available. There are some texts introducing $2$-categories in the literature, but all the references I have come across are ruled out because of the following requirements:

It really should contain the unravelled definitions of $2$-category, lax functor and lax transformation;

I do not want the word "enriched" to be used;

If the word "natural" is used, then the meaning of the naturality should be explained in full;

The vocabulary has to be consistent with current usage;

The $2$-categories have to be strict because I really do not make any use of any other objects. This condition is, however, less important than the previous ones, but it would be clumsy to write things like "for the unravelled definitions, see Leinster's "Basic bicategories" and strictify the coherence conditions".

I think the "big.list" tag is not approppriate because I am really looking for a single reference which would fulfill these conditions.