Learn English – “I will not be in these next two weeks.” meaning

There is no hard rule here. People can and do use both methods.

In casual speech people will frequently count by whole weeks, and round up. The listener will accept the ambiguity, as a precise count of days is probably unimportant.

But if there is reason to emphasize the precise number of days that have past, you can give a number of days instead.

In some instances, people will give a number of hours covering multiple days (e.g., "in the last 36 hours/48 hours/72 hours I have been doing X.") if they want to put even more emphasis on the passing of time.

The question needs context. You need to ask your friend for more clarification. Relative dates can be quite confusing. If your friend wanted to be more precise in his question, he could add "from today".

"Last two weeks from today" means the last 14 days. Whereas "The last two weeks" could have many meanings depending on context.

Consider this scenario. Today is Friday, February the 3rd. I ask my friend "What are you doing this Sunday?" which would imply the next Sunday on the calendar (Feb 5). Or I could ask "What are you doing next Sunday", which could imply next week (Feb 12). Even though I asked about "next Sunday" it's not actually the next Sunday.

If somebody asked you "Where did you work last year?" they likely wouldn't mean literally all of 2016 or even the previous 365 days. There would likely be more context to it. Consider this conversation.

Judy says

"Thanks for coming to visit me on my Birthday!", to which I reply

"My pleasure! It was easy for me to get here because my new office is just around the corner."

"Where did you work last year?"

"On the other side of town."

Judy isn't asking me what my job has been for the last 365 days, she's asking me where I worked on her birthday last year.

It is a confusing topic that native English speakers still get confused about.

In your specific scenario, it wouldn't be rude to ask him "Do you mean this week and last?" or "Do you meant the last 14 days?" or "Do you mean last week and the week before?". I personally have found it common to ask for clarification of dates in casual conversation. Legal documents and conversation where it really matters would likely have explicit dates.