The problems described in this post are unlikely to happen. There is an attempt to split ("fork") the network scheduled for August 1. The people forking will force activation of a new feature, Segwit, while the non-forkers won't. However, the non-forkers are currently planning to activate Segwit as part of a compromise plan before the deadline. If this compromise happens as planned, there will be no need to force-activate Segwit with a fork, and so no fork will happen on August 1.

Frankly, even if the compromise solution fails and the fork does happen on August 1, it will be a complete non-event. Bitcoin.org is biased as they are affiliated with people who support the August 1 fork, and so they're attempting to publicize it. However, the fork has practically zero support from Bitcoin miners or exchanges. On Aug 1 the vast majority of miners and exchanges will stay with the current network. Without significant miner support the forked network will run extremely slowly, and it will be vulnerable to several kinds of attacks. Without exchange support the forked network will not have economic value, and will quickly become irrelevant.

Although August 1 will likely not be a problem either way, there is another date that will. Around the end of October, another proposal to fork the network is scheduled, and this one is supported by miners and exchanges. What will happen then is much more murky. It will become clearer as the date approaches.