The quote in the beginning is not from the original Rogue (1980) but from the Epyx Rogue (1985). You can find the original manual here. The section "2. What is going on here?" shows quite well what the innovation of Rogue is (combining adventure with "screen-oriented" and randomness), and people started using "rogue-like" somewhere around 1982 to refer to what. Something similar to the part you quote could be written to mostly any ~1980 game.

The term as you describe here is basically meaningless and thus not helpful. "It’s a genre that can be any genre"? What do you mean by "lose and repeat", in almost all games you can try again after you lose? What do you mean by a "static starting character"? Randomized replayable games always existed, they just started being marketed as "roguelike" at some point, and there are lots of games marketed as "roguelike" which have no procedural environments, single character, or upgrades. In the roguelike communities, "roguelike" still refers to a specific genre (single-character grid tactics), and there are hundreds of such games, many of which can be played for years.

To your Doom example, you could add DRL (Doom Roguelike), which is a free Doom fangame and also a very good coffebreak roguelike. It is also very good at showing what a roguelike is. Doom and Nightmare Reaper are "first-person shooter" genre, while DRL is "roguelike" genre.