I see that article you link is only a partial list of prepositions. To get the full list, the author suggests you buy the book. So I wouldn't take anything on that page to be 100% accurate or complete. Instead I would say that, with your examples, the use of "on" and "in" is optional.
You can see from this Ngram that the use of a phrase like "on the next day" has steadily decreased since 1900 (although the total number of sources is very small, so this might be an anomaly). It may be that the modern use of "the next day" without on would sound weird to someone from the 19th century.
If you choose to use on -- not something I prefer, but perfectly acceptable -- it helps to emphasize a particular point in time.
You will arrive the evening of the 10th. On the next morning you should be ready to start training at 6 am, so we recommend you get a good night's sleep.
The contract says we will be paid on the last day of the month.
In seems perfectly natural. A phrase like "in the last few days" is common and its use steady since 1940. Again, in is not necessary, but it can help emphasize that the action occurred continuously or regularly over a period of time.
In the last few months, we've seen a steady decrease in the price of crude oil.
In the last year, I've tried to go to the gym every day.
"Over" may sound more natural than "in" or "on", when talking about a period of time:
Over the next few nights, they evaluated the volunteers' sleep ...
The Commission has talked frequently about Robin Hood over the last couple of weeks.
Over the last year I've gone to the gym every day and lost 20 kg.
When discussing something that happened, or happens, during a period of time, is very difficult to give a set of rules regarding when to use 'over' or when to use 'during'. Sometimes you must use one of them, other times you must use the other, and at other times again you can use either one of them.
As a very general rule, 'over' is generally preferred when you are talking about something that happens regularly or continuously during a period of time, although during cannot always be excluded. On the other hand, 'during' is more likely to be used if you are speaking about something that:
(a) only happened once, or perhaps a few times, during a time period, e.g. I went to the movies three times during the holidays, or
(b) something that happened at some time during a longer period of time, e.g. The Grand Final will be held during August, or
(c) something that happened at some unknown time during a larger time period, e.g. We believe the theft occurred at some time during the weekend, or
(d) something that happens during a named time period, e.g. 'during the war', 'during the depression', 'during the '40s', 'during the Elizabethan period', 'during the Clinton administration'.
However,it is also possible to use 'over' for both (a) and (c) above, but less likely for (b).
With respect to each of your sentences, I will place a + sign if I think it is acceptable, a - sign if I think it is not as acceptable, and an * if I think one answer should be preferred over the other.
Will you be home over the summer vacation? +
Use this if you mean 'Will you be home for most or all of the summer'.
Will you be home during the summer vacation? +
Use this if you mean 'Will you be home at any time during the summer'.
Over a period of ten years he stole a million pounds from the company.
+
Use this if you mean money was stolen regularly throughout the ten year period, totaling a million pounds. i.e many small thefts
During a period of ten years he stole a million pounds from the
company. +
Use this if you mean that at some stage during the ten year period he stole a million pounds. i.e. a single theft, or possible a few large thefts.
Wages have fallen by more than twenty percent over the past two
months. +*
Wages have fallen by more than twenty percent during the past two
months. +
Either is possible, but I prefer 'over the past two months'. Generally, this sentence would make more sense with a longer time period (eg six months).
Plants need to be looked after and protected over bad weather. -
Plants need to be looked after and protected during bad weather. +*
I can't say why, but 'protected during a time period' sounds much better to me than 'protected over a time period'.
American business in Britain over the 1950s grew much faster than
British business. -
American business in Britain during the 1950s grew much faster than
British business. +*
'The 1950s' is a named time period.
Over his lifetime he was relatively unknown. -
During his lifetime he was relatively unknown. +*
'During his/her lifetime' is the commonly accepted form, although 'over his lifetime' is not entirely unknown.
Many strikes over the last few years have not ended successfully. -
Many strikes during the last few years have not ended successfully. +*
Strikes tend to occur sporadically over a period of years, rather than continuously, so I prefer 'during'.
Best Answer
Last week, month etc. is the week, month just before this one. The last week, month etc. is the period of X days up to the moment of speaking.
Sometimes the last means the last in a series. If you want to say that something happened during this period, you say that it happened in the last three weeks or during the last three weeks.
If you don't say the number, you can say
Or use recent instead. For example, you can say
The past+ a noun is used to refer to the period up until now. ("their activities over the past two years").
The sources are Longman Exams Dictionary, M. Swan PEU, Collins COBUILD English Usage
I think it is clear that you should use prepositions in your sentences."For"(to say how long),during,in,over (something goes on within a period of time) would do.