Yes, this is important. It depends on whether your buyer is a strategic acquirer, a private equity fund or simply some party that wants to run the business 'as is' because they want a new challenge.
I'm going to hazard a guess that the party you have in mind is in the first category.
To be blunt: I don't think it is possible to draw a line between 'acquirer is acting in good faith' and 'acquirer is trying to get critical information'. Unless you feel that you are well protected, that your business is hard to duplicate and that you have no secrets that you do not want them to learn you will be taking risk, sometimes a lot of risk.
The best way to mitigate that risk is to ensure that only independents have access to your internal data prior to the deal and that those independents are beholden to you and to you alone. This may cause the deal to break off, but that too is a risk that you may have to take.
Ultimately, the question is one of trust, and the thing you have to ask yourself going in with the items that you are going to disclose 'am I going to regret doing this if the deal does not go through'. One way to get more comfort is to negotiate a break-off fee that gets parked in escrow conditional to the acquiring party backing out of the deal. That amount should be carefully calibrated to allow the other party to back out with some grace and for you to have sufficient compensation for anything that they might have learned that they can use to their advantage (even if indirectly).
Get yourself a good corporate finance advisor for strategic acquisitions, set up a carefully audited dataroom and just as much as you should look at the acquirer you should also look at the parties doing the DD (especially since you won't be paying them). If things are not to be disclosed to the acquiring party make sure your deal team knows exactly what these items are so nobody goofs and accidentally hands over the crown jewels, keep in mind that this is all 'upside' to the acquirer so nobody will get mad with you for keeping that to yourself. If you have trade secrets do not allow these to be handed over during DD, only post acquisition.
What is your opinion on how to do DD on the buyer ?
In particular to address the question on whether buyer is fishing for insider knowledge, and not serious on their interest.
The article had 1 sentence on the subject, but didnt go into any details.
From experience, this is a critical point, possibly the most important point to establish pre DD.