This is a really good question for the current working environment, where work and home are colliding for many of us, in unpredictable ways. The best answer for you will depend on multiple things, that are not clear.
You could try to solve this with a social solution - chatting with your boss, reminding him that this is your personal laptop, that you do have a social life outside of work, and that there are things about your personal life that you do not want to share with everyone at work. Explain how it makes you feel when you are being expected to open up folders that leak the names of personal files to others in the meeting, that could include private medical details or aspects of your love life that you don't wish to make public knowledge. Making video recordings of these meetings only makes this worse. Most bosses will understand and adapt. A good boss would have picked up on your unease and adjusted the work style. A really good boss would not have placed you in this situation in the first instance, would have thought through the implications of using your home computer for work - and would have had a plan to reduce the risk of this.
There are technical steps that you could use to place some isolation between home and work life. This is probably a good idea to do, even if one of the other options is the main route that you choose to use. This job may not last as long as this laptop, and they will want you to remove any work stuff when you part ways. I'd probably do something as simple as creating a seperate user on your laptop for work - this way most folders are completely seperate from your personal folders.
Legal / Employment solutions
This is probably the least likely approach to get a good, harmonious outcome. Depending on where you live in the world, there may be some legal protections for personal privacy, right to a family and personal life outside of work and also for protection from unreasonable harassment.
Your employer should be providing you with the essential equipment required to perform the role, the fact that they haven't doesn't automatically mean they can require you to use your personal kit. The fact that personal dating history has been exposed against your wishes, and that this made you feel uncomfortable crosses the line of sexual harassment in some countries. (Its sexual, it is unwanted and it made YOU feel uncomfortable. The intent of your manager is irrelevant). Let your employer's HR department know that the lack of your work computer is an issue that the company needs to solve urgently - and they also need to put in some safeguards into the working environment for you whilst the computer remains in transit.