Are there different kinds of invalid arguments? Here are two invalid arguments from my semantics class:

*

*$$\begin{array}{rl}
    & p \lor q \\
    & \neg p \\
    \hline
    \therefore & \neg q
  \end{array}$$

*$$\begin{array}{rl}
    & p \to q \\
    \hline
    \therefore & \neg (r \to q)
  \end{array}.$$
You can't make Argument 1 valid by adding more premises, since you can't both satisfy the premises and the conclusion, since $\{p \lor q, \neg p, \neg q\}$ is not simultaneously satisfiable.
But you can Argument 2 valid by simply adding $r \land \neg q$ as an additional premise.
Are there different names for invalid arguments, like Argument 1, that are logically inconsistent, and those, like Argument 2, whose premises are not strong enough to support the conclusion?
 A: 
invalid arguments, like Argument 1, that are logically inconsistent

To be accurate: argument 1 $\Big((p∨q)∧¬p\;\to\;¬q\Big),$ although invalid, is a contingency and thus is logically consistent; what you mean (and said earlier) is that its premises and conclusion are inconsistent with one another (i.e., aren't simultaneously satisfiable).

You can't make Argument 1 valid by adding more premises, since you can't both satisfy the premises and the conclusion

Actually, argument 1 immediately becomes valid by adding any contradiction as a premise.
On the other hand, this is correct:

*

*If the premises and conclusion of an invalid argument are not simultaneously satisfiable, then adding any premise cannot make it sound (valid with premises all true).

A: An argument (text) that has a missing (deliberately suppressed) constituent to be valid is called elliptical. So, Argument 2 is of the sort elliptical (if it is not a mistake and the context makes this clear for the audience) expressed in the contemporary symbolism.
An enthymeme is an elliptical argument (text) in which one of the premisses or the conclusion is not stated.
The following examples for enthymeme are from Bachhuber's Introduction to Logic (p. 160):
\begin{align*}Major: &\text{ What is spiritual is immortal.}\\Minor: &\text{ But the human soul is spiritual.}\\Concl: &\text{ Therefore the human soul is immortal.}\\\\1. Minor:&\text{ The human soul is spiritual}\\Concl:&\text{  and therefore immortal.}\\\\2. Concl:&\text{ The human soul is immortal}\\Minor: &\text{ because it is spiritual.}\\\\3. Major:&\text{  What is spiritual is immortal.}\\Concl:&\text{ For this reason the human soul is immortal.}\\\\4. Concl:&\text{  The human soul is immortal,}\\Major:&\text{  since whatever is spiritual is immortal.}\\\\5. Minor:&\text{  The human soul is spiritual,}\\Major:&\text{  and whatever is spiritual is immortal.}\end{align*}
Terminological note: Since the origin of the term 'enthymeme' is Aristotelian syllogism, some authors restrict its usage to syllogism only. Some authors use the "elliptical argument" only in case that a premiss or conclusion is missing, in contrast to any missing, but implied, part of an argument (text).
It is worth a remark that elliptical arguments are admissible only if asserted on some purpose, not by mistake, otherwise they are merely invalid as Argument 1. In mathematical logic, they are formally inadmissible in general, and mostly studied in the provinces of argumentation theory, rhetoric and philosophical logic.
