Differences between False Equivalence, Extended Analogy, Weak Analogy

This is how I understand them and I'm gonna stick to mathematics to give examples:

In false equivalence, there are two statements, possibly sharing a characteristic that gives the appearance of equivalence but they're actually not.  

In extended analogy I see it as an attempt to use transitivity of some relationship, but that relationship is not that strong enough to warrant the move.

In weak analogy, we provide an analogy in attempt to make it an apt analogy, but it is not apt. This one may or may not be a result of the previous two.

Example 1 (false equivalence):  
f and g are continuous functions. But g is differentiable also. Therefore, f is differentiable.

f and g have a shared property giving them an apparent equivalence. But we cannot say that f is differentiable if g is.

Example 2 (extended analogy):
a and b are divisible by 2. b and c are divisible by 3. Therefore, a is divisible by 3.

There is an assumption that divisibility is transitive independently of the divisor. The "analogy" has been extended.

Example 3 (weak analogy):
The number of ways to select 3 letters from the word "weak Análogy" is equal to the number of 3-element lists out of a 12-element set (including a space).

This one is more complicated but here's the point: I believe that the field of combinatorics deals a lot with one-to-one correspondences, and we can see them with analogies. The correct solution is to say that choosing 3 words, all distinct and including the space, is similar to choosing a 3-element subset from a 12-element set because the order doesn't matter. The other analogy does regard order.

Now, I think that example 3 is not a false equivalence nor an extended analogy because there is no one-to-one correspondence and I believe that's all it takes when it comes to counting.

Weak analogy is taking two 'analogues' (X and Y), and saying that because X is like Y and X has property P, then Y also has property P. The problem is that X and Y are too different. In this fallacy, you're not saying X and Y are 'equal', you're just saying they have a certain common property.

Extended analogy is interpreting the use of an analogy as a statement of equivalence between X and Y. Basically, if someone says that X has property P, and Y also has property P, it is fallacious to interpret that as being "X = Y".

False equivalence is claiming that because X and Y share a property P, then they are equal. It is the informal version of fallacy of (the) undistributed middle.

So yeah, they're similar - they all have something to do with similarity of the comparatives. As for whether it's worth differentiating them, I think it could be justified.

There are significant enough differences to keep them separate. Besides, as much as I would like to think it, I am not the arbiter of Fallacies :) These are widely known the Internet.

I see a couple of benefits in keeping them separate:

1. (and perhaps the weaker of the two) – each of the three stems from a slightly different breakdown in logic.

2. if we assume that the three fallacies were committed innocently and if we further assume that the aim is to search for a way to improve the arguments in which these fallacies appeared, the solution would be different.

In the case of false equivalence, we'd need to find examples where the background and details are closer to being the same.

In the case of extended analogy, we'd need to show how there are similar connections among all of the subjects.

In the case of weak analogy, we'd need to find examples that are closer together.

While the basic issues may be similar, the way to improve the argument is different so, for me, it's worthy of separating the three