You can not rely on other games to say your Internet is fine as they are all different and the Destiny games are VERY sensitive as they use a P2P host system (in fact D1 and D2 make a great early warning system to when I start to have issues). You are having data loss issues. Reset your router and then your systems cache to refresh your Internet connection. Run a Internet diagnostic from your end looking for issues. Do not use WiFi but a link cable and if you need to use WiFi use signal boosters. If needed contact your ISP and tell them you are having data loss issues and PUSH them to investigate and help you as they only look to see that you have Internet at the speed you pay for not if that signal has issues.
I thought that a PS4 had everything you needed to run a Internet diagnostic built in as standard just like my XB1.