A question regarding converse errors. Consider the following argument form -
$$ \text{If}\ p\ \text{then}\ q$$
$$p$$
$$\therefore q$$
I can see why this argument form is valid by constructing a truth table. But I'm not sure whether I understand it intuitively. If I were to explain why this argument form is correct without using a truth table, my explanation would be as follows -
$\text{If}\ p\ \text{then}\ q$ means that if $p$ is true, then $q$ is certainly true too. So if $p$ is true (as given by the minor premise), by the definition of a conditional statement, $q$ is true too.
Additionally, If I were to explain why the following argument form is incorrect without using a truth table -
$$ \text{If}\ p\ \text{then}\ q$$
$$q$$
$$\therefore p$$
then my explanation would be as follows -
$q$ therefore $p$ means $q \rightarrow p$. This is not the same as $ \text{If}\ p\ \text{then}\ q$, which is why the above argument form is invalid.
As you can probably tell, I'm viewing the major premise and conclusion as "the major premise implies the conclusion."
$\large\textbf{Questions}$
Do my justifications as to why modulus ponens arguments are correct and why converse error arguments are incorrect make sense?
 A: Your first explanation is fine.
The explanation for why the second pattern of reasoning (which is typically referred to as the Fallacy of Affirming the COnsequent)  is really not. Just because some argument does not follow the same pattern as Modus Ponens does not mean that it is not valid.
In effect, you are making the following argument:
$$\text{If an argument follows a Modus Ponens pattern, then it is valid}$$
$$\text{Here is an argument pattern does not follow a Modus Ponens pattern}$$
$$\text{Therefore, here is an argument pattern that is not valid}$$
The abstract form of your argument therefore is:
$$ \text{If}\ p\ \text{then}\ q$$
$$\text{not } p$$
$$\therefore \text{not } q$$
This pattern is often referred to as the Fallacy of Denying the Antecedent, and (as the name says) it is not valid!
Why?
Well, as other commenters have already pointed out, a good way to show that an argument is invalid (without using a truth-table) is to come up with a counterexample:  some scenario where the premises are true but the conclusion is false. So, let me do that for the above argument.
First, consider yet another argument pattern called Modus Tollens:
$$ \text{If}\ p\ \text{then}\ q$$
$$\text{not } q$$
$$\therefore \text{not } p$$
Note that the pattern of this argument is not the Modus Ponens. So, by your argument, it is supposed to be invalid. But Modus Tollens is valid: Without using  truth-table, I can explain that by pointing out that if $q$ is true whenever $p$ is true, then if we know that $q$ is not true, then $p$ cannot be the case, because if $p[$ were the case, we would have $q$, which we don't.
OK, so Modus Tollens is a counterexample to your earlier argument pattern: we do have that if Modus Tollens would be of the form Modus Ponens, then it would be valid. We also have that Moduss Tollens does not follow the Modus Ponens pattern. But, we do not have that Modus Tollens is not valid.
Finally, note that we can also point to Modus Tollens as a counterexample to the Affirming the Consequent pattern of argument. Again, we can say that if Modus Tollens would be of the form Modus Ponens, then it would be valid. And we also have that Modus Tollens is valid. But we don't have that Modus Tollens follow the Modus Ponens pattern.
