From what I can deduce, the problem seems to be that the tube bulges through the rim tape and into the spoke holes. Apparently this stretches the tube too much, or it gets cut by the edge of the spoke holes, causing the puncture. There is no way to patch this kind of punctures as the cut on the tube is too big to patch.
To try to remedy this, I have placed 2 layers of rim tape this time, to try and prevent the same problem from occuring again. Wide rim tape cannot be used, as this will fill up too much of the space on the rim, making it very difficult to fit the tires onto the rim.
Without rim tape
First layer of rim tape. This is the new rim tape I got from MyBikeShop.
The original Wheelsport rim tape. It is not too bad, but can't seem to withstand high pressure very well. This goes on as the second layer of rim tape.
Another reason for this might be the tubes that I am using. For the Wheelsport wheels, which have a 30mm profile, I need a tube with a longer valve, in order to pump the tubes. The Kenda tubes seem tougher/thicker but has a valve that is too short. The Schwalbe ones have a longer valve which is more suitable for these wheels. However, the Schwalbe tubes that I got from MyBikeShop are the Extra Light SV6A, which are the only ones that they have, and they seem to be thinner and lighter than the Kenda ones. Maybe that is why these thin tubes are more prone to punctures?
Anyway, with the double layer of rim tape, it seems unlikely that the same problem will occur again. The rim tapes that I used are rather narrow and thus do not affect the seating of the tire onto the rim. Shall monitor the situation for a few days and hopefully nothing happens!