Colorize takes from the original image only the brightness variations. Hue and saturation are determined by the sliders. Leave colorize unchecked and you can shift the hue and saturation as you did in your 2nd image. The lightness slider affects in the same way, no matter is the colorize mode ON or OFF.
You can use Image > Adjustment > Hue/Saturation with hue shift +89 to get the shown effect without inserting an adjustment layer. It can be recorded into an action to apply the same to several images. Learn how to use actions to repeat a set of adjustments for a folder of images.
Not asked: Full white and black do not take color when one applies colorize. You must reduce contrast to make them grey. That's a limitation of the RGB color system.
If you are after more ways to get weird color changes you can try to mangle the R,G and B curves separately after opening Image > Adjustment > Curves:
The result is surely difficult to predict, but generally it reduces contrast. You can later increase contrast or saturation to make it bright again.
As a third method you can try Gradient map. It's is a popular way to change the brightness scale to a predefined series of colors:
Using adjustment layers for searching a good effect is recommended. The image below them stays intact and it can be changed. And you can save them for future use if you save the image as PSD.