The Set Key Animation Mode won't add the keyframes automatically. Which might be a good thing when you forget you are in animation mode and start moving things around in your scene and your changes won't be saved as permanent, but as an animation. (Yes, it did happen to me very often).
But, it can also be a risky business if you forget to add the keys yourself. So how does it work?
Click on Set Key. The button turns red, the viewport gets highglighted in red, just as before. Now, move the time slider to frame 0, and this time, you need to press Set Keys , the big key icon right next to the 'Set Key' toggle. You'll see that now a keyframe icon has been added in your timeline, with the current coordinates of your object. Move your slider to frame (insert the frame you want here), make a change to your object (rotate it for example) and don't forget to press Set Keys again. If not, your changes will be discarded and your objects rotation (in this case) will go back to the one in frame 0.
Yes, you can still edit modifiers.
One more thing, there is a toggle called Key Filters . The pop-up will let you choose what you actually want to animate and you can single that out, or ... choose everything from that list. Makes it easier if, for example, you only want to animate rotation. In case you change the position by mistake, that won't be taken into account.