The infinitive and dictionary form are the same thing. Words (action and descriptive verbs) in this form will all end in 다. It is a base form. That is, other conjugations are all created from these forms of the word. It is not used in speech or writing. Another form ending in 다, the plain form, is however.
Here is the infinitive/dictionary form: 먹다
Plain form: 먹는다
Yes, it is confusing because all resources use different terms. It is not that important to know the names of each in my opinion, at least not in English since everyone uses different terms to describe the same thing. There are standard Korean terms for each though.
More importantly is to know the ending and what it is used for instead of the name.
먹어 is 반말 ("half words/language"), a very informal way of speaking generally used for animals, friends, children, family members, those younger than you (basically it all depends on your relationship to the speaker)