Is there a terminology contradiction about whether the conjugate of a strong acid is a "weak base"?

(Please note, I am not simply asking, "Is the conjugate of a strong acid a weak base?" I'm asking about the contradictory ways those terms seem to be used.)

I was revisiting strong/weak acids/bases to help some high school chemistry students and found a lot of apparent contradictions in the way that sources use the terms. One is whether the conjugate acid of a strong base is considered a "weak acid".

I understand that the strongER the base, the weakER the conjugate acid, and all sources use the relative terms consistently. The contradiction is in the use of the absolute terms "strong" and "weak".

There seem to be two conflicting conventions in use. In Convention I, the conjugate of a strong base is indeed called a weak acid (whereas weak bases and weak acids can also be conjugates), so you have these conjugate pairings:

\begin{array}{cc} \hline \text{acid} & \text{base} \\ \hline \text{strong} & \text{weak} \\ \text{weak} & \text{weak} \\ \text{weak} & \text{strong} \\ \hline \end{array}

In Convention II, the conjugate of a weak acid is a weak base, and vice versa; the conjugate of a strong base is too weak to be considered an acid, and the conjugate of a strong acid is too weak to be considered a base:

\begin{array}{cc} \hline \text{acid} & \text{base} \\ \hline \text{strong} & \text{not a base} \\ \text{weak} & \text{weak} \\ \text{not an acid} & \text{strong} \\ \hline \end{array}

You can find examples of both just by googling "conjugate of a strong acid", "conjugate of a weak base", etc. Many sources complete the sentence "conjugate of a strong acid [or base]" with "is a weak base [or acid]" (Convention I). On the other hand many sources complete "conjugate of a weak acid [or base]" with "is a weak base [or acid]" (Convention II).

I just want confirmation different sources really are using these terms in ways that contradict each other, so I can pass that on to my students and move on.

So, which of the following is true?

1. 
   

   There is no contradiction; I'm reading it wrong. (This seems unlikely since you can literally google "conjugate of a weak acid" and get what seem to be multiple contradictory answers.)

   
   
   


2. 
   

   One usage of these terms is correct, and the sources that say otherwise are using the terms incorrectly. (In particular, the more reliable sources seem to skew towards saying that the conjugate of a weak acid is a weak base, i.e. Convention II. Maybe the sources that say "the conjugate of a strong base is a weak acid" are wrong, and they're just garbling the (correct) statement that the strongER the base, the weakER the conjugate acid.)

   
   


3. 
   

   There is no agreed-upon definition of "weak acid" and different sources use it in contradictory ways, but real chemists don't care because they just use Ka values.

   
   

To avoid horrible confusion, please be clear in your answer if you're voting for #1, #2, or #3, or something else not listed here.