Well, this is a bigger mess than I had anticipated, but I'll start by welcoming Baja to ATP. Thank you for finding this ball of confusion and bringing it to our attention.
From a cursory scan of the online descriptions of these three (or two?) plants, I can find that Logee's is selling a cultivar called 'Firesticks,' but its description states that "The red stems often show shades of pink, orange or yellow," so it would seem that designations by color could be guesswork at best.
In most other cases on the Web, the three are treated as one plant, just as they are in our database. The most common treatment is to call the cultivar 'Rosea' and to list Firesticks and Sticks on Fire as alternative names for the 'Rosea' cultivar. This agrees with our database, where the cultivar is 'Rosea,' but Firesticks is listed as one of its common names, and Sticks on Fire is listed as an also-sold-as name.
Pencil Cactus (Euphorbia tirucalli 'Firesticks')
Here's one of the sources that probably was used as a basis for our database listing.
I'm quite frankly stumped because I find no source showing more than one plant, although each source might show it under a different name. Baja, can you find me a source acknowledging the existence of more than one plant?
Also, Baja, I should clear up a misconception. The photos in our dataspace species entries are not meant to be confined to the "wild" type. People add their photos to the species entry when they don't know the name of the cultivar. In that sense, all of the photos currently in the E. tirucalli entry do belong there. Any member who recognizes one of these photos as a specific cultivar can propose that the image be moved to the cultivar entry. This is easily done. Click on the photo to enlarge it, scroll down to the "Actions" block, click on "Move this image to another plant," type the name of the other plant in the box provided, and submit the proposal. When the proposal has been approved by an admin or a moderator, the image will be moved.