I don't think we have any real disagreement except for terminology. A compiler translates one formal language to another. If you define the parse tree to be a language, your definition of "language" is loose enough that every data structure is a language, and that makes every program a compiler. Saying that a language is "compiled" in the old-school sense really means that it is "compiled to machine language," but as you say, the distinction is mostly unimportant these days.
Contrary to some other languages (e.g. shell, TCL, awk, makefile), Perl has clearly a compile phase and a run time phase.
All of those languages have to be parsed, so they clearly have a "parse phase." The thing about Perl is that it lets you execute Perl code during the parse. Off the top of my head, I can't think of any other language that lets you do that.