Learn English – Is ‘I will definitely be living away from the shop’ correct

In most contexts, I am going to [verb] and I will [verb] are interchangeable. Sometimes the former may place more emphasis on the fact of your current intention/expectation, where the latter emphasises the future action.

There are some contexts where the difference is clear, and this may have some bearing on why OP thinks there's a planned/spontaneous distinction (usually there isn't). Say you're round a friend's house watching a football match on TV, and at half-time the beer runs out...

You: "I'll nip down the shop and get some more beer."

Friend: "It's raining - I'll find my umbrella for you."

If you reply "Don't bother - I'm going to use my car", the implication is you had already decided you were going to use the car before you first said you'd get the beer. But if you say "Don't bother - I will use my car", this implies you just made that decision in response to what your friend said.

@Peter Shor mentions another context ("It's going to/It will bite!") where native speakers often make a distinction. In that case, going to usually signifies immediate danger (it's just about to bite), but will can just mean it's bitten others before, and will/may bite you soon if you're not careful (effectively, the same as "Careful! It bites!" where present tense indicates "habitual" action).

Kosmonaut's "Let's say that tomorrow you will walk your dog" isn't really relevant to the current issue. It could just as well have been "...tomorrow you are going to walk..." or even just "...tomorrow you walk...". They all mean the same, and in that context there's no real reason to prefer one over another.

To summarize what follows, the distinction between your different intents is defined less by the verb tense and more by the precision with which you specify future plans or appointments.

If I've already booked a flight or have made an appointment, respectively, it's a (personal) arrangement, thus present continuous:

Next month, we're spending our vacation/holidays in Canada.
I'm seeing the doctor tonight.

This is correct and acceptable usage, whether you've encountered it personally or not. The implication here is that plans have been made, whether plans for travel or an appointment, and you have every intention to follow those plans.

I would use present tense, though, when I give the exact time as I give the exact schedule. Or is present tense wrong altogether?:

We spend our vacation/holidays in Canada from September, 1st to the 12th.
I see the doctor at 6 o'clock.

What looks like simple present tense here is actually the habitual present. It reads that you always spend your vacation/holiday in Canada from September 1 to September 12, or that you regularly see the doctor at 6 o'clock (perhaps you're married to this particular doctor).

If I haven't taken any actions but it's decided, it's my/our intention, thus going-to-future:

Next year, we're going to spend our vacation/holidays in Canada.
I'm going to see the doctor soon.

Actually, this has similar meaning as your first example, but it's the lack of precision regarding the time frame ("next year" and "soon") which indicates that you're still in the planning phase.

We're going to spend vacation/holiday in Canada from September 1 through September 12. I'm going to see the doctor tonight.

There are, of course, two verbs here: "am/are/is going" and "to spend/see". The use of "am/are/is going" means that you have the intent of pursuing completion of the infinitive ("to spend/see"). As above, it indicates that you are well within the planning phase and the precision of the dates indicates how far along you are into the planning phase.

If I/we haven't made any decision, let alone take any actions, it's a prediction/opinion/hope, thus will-future:

(I think,) We will spend our next vacation/holidays in Canada.
(I think,) I will go to the doctor.

Actually, saying "We will spend" or "I will go" gives it a particular firm feeling, so "I think" would be mandatory to soften this. As with "We're going to" or "I'm going to", it indicates that you are in the planning phase and the generality of the time frame indicates that, while the destination has been decided, the time has not.

As mentioned above, prefacing this with "I think" softens it somewhat but, believe it or not, does so in different ways.

I think we will spend our next vacation/holiday in Canada.

With the disagreement in number, i.e., "I think" and "we will", this implies a personal wish regarding the following: "I would like us to go to Canada, but I don't know if the family does."

We think we will spend our next vacation/holiday in Canada. I think I will go to the doctor.

Now that the number matches ("We think we will" and "I think I will"), this states a desire to do something while indicating little progress in the planning process.

Another reason for will-future is when I make a promise, or a spontaneous decision:

Yes, we will spend our next vacation/holidays in Canada.
Yes, I will go to the doctor.

In the right context, this is correct. Of course, if you're spontaneously deciding on the destination of your family's next vacation/holiday, you should be sure that the family will go along with it. Otherwise, you might say, "Yes, I'll encourage the family to spend our next vacation/holidays in Canada," or, "Yes, I'll propose that we spend our next vacation/holidays in Canada."

This is also the speech you might use to placate someone who is nagging you to do something. You'll get mixed results because, while you're using the firm "I will" or "we will", you're not providing a precise time frame, so it has a balance of firmness and indicates a lack of progress into the planning phase.