I just got a new S4 phone as well, and was trying to figure out the same issue you're having. From your screenshots above, it looks like we have the exact same settings (and apparently no way to make the settings that other people are referring to). I haven't tried the SwiftKey app yet, but I think I've figured out how to use the Samsung keyboard to do what I need. It might seem like a lot of steps, but if you do it enough times it gets faster.
Here's how I add/delete things to my dictionary.
(Btw, in the Samsung keyboard settings, I've got "predictive text" and "auto replace" turned ON):
To Add a Word (you can delete later using steps further below):
1. Type a word that you know is misspelled. For example, type "Sansung" instead of "Samsung".
2. When you get to the end of the word, DON'T type the space bar.
3. Instead, click somewhere in the middle of the word. For example between the s and u, as long as it's in the middle of the word.
3. Then click just after the last letter of the word (just after the g). In the word suggestions, you should now see "Sansung" as an option. Select that suggested word, and the new word should now be in your dictionary. (Test it by trying to type the word again, and it won't auto-correct it if it's been saved to the dictionary).
To Delete (Remove) a Word from the dictionary:
1. Type the word Sansung again (the misspelled word). If it offers Sansung as a suggestion, go to step 3.
2. If it doesn't offer it as a suggestion, click the space bar, and it won't auto-correct. Then click somewhere in the middle of the word, and it should then offer "Sansung" in the suggestions.
3. When you see the word in the suggestions bar, press and hold over word you want to delete (within the suggestions bar, Not within the main text).
4. You should then get a popup window that asks if you want to Remove the word from your "learned words".
5. Click OK to remove the word.
Btw, the Swype keyboard settings have a section called "my words" with an option to "edit my dictionary". But none of my custom words are being saved in the Swype dictionary, so they must be going to the Samsung keyboard dictionary (which I can't find a way to access).