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Going straight to the dictionary, we can see that:

Consecutive means: something following something else without interruption.

Some examples:

1. He did 100 consecutive push-ups (1, 2, 3, ..., 99, 100)



2. 
   

   He scored 3 consecutive goals (meaning that a player scored 3 goals without any other team-mate scoring in between)

   
   
   

   2.1 Ronaldo scored in min. 25, 50, 75. He scored 3 consecutive goals. Had somebody else from his team scored anywhere between 25-75 his goals wouldn't have been consecutive.

   
   



3. He had to take his medicine for 14 consecutive days (2 weeks straight, without interruption).

Sequential means: following a particular order.

The above doesn't mean without interruption though. It just has to follow a predictable pattern.

Some examples:

1. Computer instructions are always executed sequentially, the order being top-down as they are encountered in the program.




2. The Fibonacci sequence: 1, 2, 3, 5, 8, 13, 21, ... is a sequence where each number is the sum of the preceding two numbers.




3. I described the events sequentially, but in reverse order, starting with the conclusion.




4. We were given a list of sequential numbers comprising even multiples of 10. (20, 60, 100, say every 4 multiples)




5. He was used as a defender every 5 games for 6 sequential World Cups (this is a case where you might substitute sequential with consecutive but it's also correct to interpret this as say every 2 World Cups, or every 3 World Cups is it not?). Without knowing further information this is ambiguous just like biweekly which can mean both every 2 weeks and twice a week.




6. They were using a sequence of numbers comprising multiples of five. Note that this can also be interpreted as consecutive since it doesn't specify any further details about the multiples.

So all in all I've seen folks using these 2 terms interchangeably. While this might be true it's not always the case. They can be used interchangeably when the elements that they're describing are immediately following one after another but not otherwise.

Is the above correct or did I misunderstand anything?

OP's understanding of the words 'sequential' and 'consecutive' is correct, but the question "Is “consecutive” a subset of “sequential”?" is a little too mathematical in construction. Ordinary word definitions don't follow mathematical concepts like that. It's probably true that in most cases something that is consecutive is also sequential, but that doesn't really explain word usage. Describing something as "sequential" puts on focus on the fact that what is being described is done in some particular order, as opposed to doing things out-of-order or simultaneously. No one would use the word "sequential" to distinguish between consecutive and non-consecutive.

"Consecutive", on the other hand, puts the focus on the fact that there is no interruption. In shorthand, to describe something as "sequential" you mean to focus on the order; to describe something as "consecutive" you mean to focus on the lack of interruption.

As for the examples, I would say that #5 for "sequential"—"He was used as a defender every 5 games for 6 sequential World Cups"—is not a particularly felicitous use of the word. I agree that "consecutive" would make more sense here—you wouldn't use the word "sequential" without further explanation if you meant something exotic like the suggestion of every 2 or 3 World Cups. If that scenario was meant, then it would have to be explicitly mentioned. The fact that such a scenario could be described as "sequential" is not sufficient to use the word without further explanation.

The only other thing to consider is that sequential doesn't even necessarily mean monotonic. If you said that the sequence was "5 1 3 2 99" then following that sequence is no more or less sequential than, say, skipping every 2.