The mechanism of the butadiene + ethylene Diels-Alder is not wrong as drawn. It shows the concerted movement of electrons from the diene to dienophile resulting in cyclohexene. This correctly describes the bond formation. It is unusual to see the parent Diels-Alder reaction explained through those minor resonance contributors. We often draw mechanisms through minor resonance contributors to explain aspects of reactivity.
However, there are several problems with describing the Diels-Alder reaction this way. First, this tries to force the Diels-Alder reaction into the paradigm of "polar" reactions, i.e. nucleophile-electrophile. Although we learn some reactions of dienes and isolated alkenes as nucleophiles (electrophilic addition of H-X, X-X), there are essentially no reactions of electronically neutral dienes/alkenes as electrophiles. One of the key lessons of the Diels-Alder reaction is that it introduces a class of reactions that are "non-polar" (related question), the pericyclic reactions.
The second problem is how to handle substitution. The Diels-Alder reaction can create four stereocenters, which suggests that upto 16 stereoisomers can result. However, that is not what is observed. There are only specific stereochemistries of products that result from specific stereochemistries of reactants: the Diels-Alder reaction is stereospecific. Without invoking molecular orbitals, explaining this is impossible. One can learn the rules for stereochemical outcomes, but the rationale relies on the interaction of MO's.
The addendum to the question gives some indication of what the instructor was thinking when introducing the Diels-Alder reaction. Although the Diels-Alder reaction is not a traditional nucleophile-electrophile reaction, it does follow frontier molecular orbital theory: one of the component's HOMO interacts with the other component's LUMO. Typically, the dienophile provides the LUMO, which is the traditional role of the electrophile. An alkene conjugated to an electron withdrawing group (such as a carbonyl) is more electrophilic. It's also a more reactive dienophile. So there is a parallel between electrophilicity and dienophile reactivity both explained by having a lower LUMO.
The same is true with respect to the diene component. Typically, the diene provides the HOMO, which is the traditional role of the nucleophile. A diene substituted with an electron donating group (such as a methoxy group) is more reactive toward dienophiles and electrophiles.
So while the Diels-Alder reaction is non-polar, it can have polar character. It turns out that drawing resonance structures of the reactants is very useful for predicting the regiochemistry reactions with polar components. The major product will result from bond formation of the most nucleophilic end of the diene with the most electrophilic end of the dienophile (normal electron demand DA). Those resonance structures are simply substituted versions of what the instructor started with.