"Be going to" is Future Simple?

According to this source the form "going to" is really a form of the future simple, this latter being called so, apparently, in opposition to the future continuous.

It is a construction for expressing the future that has been used in the naming of an aspect of the future as the going to future (Wikipedia) as opposed to the will future. The form is the going to form.

The going-to future is a grammatical construction used in English to refer to various types of future occurrences. It is made using appropriate forms of the expression to be going to.[1] It is an alternative to other ways of referring to the future in English, such as the future construction formed with will (or shall) – in some contexts the different constructions are interchangeable, while in others they carry somewhat different implications.

(abridged)
There is no clear delineation between contexts where going to is used and those where other forms of future expression (such as the will/shall future, or the ordinary present tense) are used. Different forms are often interchangeable. Some general points of usage are listed below.

• The going-to future is relatively informal;
• The going-to form sometimes indicates imminence, but sometimes does not; and it sometimes indicates intention, but sometimes does not (compare "It's going to rain", which expresses imminence but not intention, and "I'm going to visit Paris someday", which expresses intention but not imminence).
• The "will" future is often used for announcing a decision at the time when it is made, while going to is more likely for a plan already in existence: compare "All right, I'll help her" and "Yes, I'm going to help her".
  The will future is used more often than going to in conditional sentences of the "first conditional" type: "If it rains, you'll get wet" (although going to is also sometimes found in such sentences).
• In some contexts the going-to form can express unconditionality while the will form expresses conditionality ("Don't sit on that rock, it's going to fall" means it's going to fall regardless of what you do, while "Don't sit on that rock, it will fall" means that it will fall conditional on your sitting on it). But in some contexts (particularly with "future in the past") the reverse can be true ("After 1962 ended, I would be a star" unconditionally describes what subsequently did happen, while "After 1962 ended, I was going to be a star" describes only intention).

I am going to Paris' is present continuous as a future form with 'going' used as the main verb; it's not future simple with 'going to' as long as 'Paris' is not a verb).

This seems like gobbledegook to me. The more I read it, the less it makes sense.

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My take on this

"I am going to Paris" can either A. be present continuous or B. it can be a prediction (there is no formal future tense in English).

A.

Police officer: Where are you going? (request to know your current activity)

You: I am going to Paris. (describing your current activity in the present)

B.

Friend: Where are you going for Summer next year? (request for future plans)

You: I am going to Paris. (prediction of your future activities)

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Answer

"We are going to stay with our family in Aberdeen."

I agree with your decision that it is "Present Continuous used as a future form".

The key is whether “going” is used in the sense of travelling.

If it is (as in your “going to Paris” example), then it falls under your ‘Continuous as Future’ category.

If it isn’t, then “(be) going to” is used a synonym for “will” or “shall”. They are not taking about motion - they are stating an intention. In that case, it falls under the same category as “will” and “shall”, namely, ‘Future Simple’.