It is entirely possible that the aircraft had a fuel leak. It can sometimes take up to thirty minutes to establish that a leak actually exists. Although fuel is normally stored in the wings and wing box fuselage area, it is delivered to the engines, as such there is a lot of pipework around the engine and engine strut area. Although quite rare, the engines are obviously a prime location for leaks to occur if and when they do. Once a leak is confirmed, it can often be stopped or reduced by shutting down the relevant engine. That might still result in spray being observed, or some pooling after the aircraft has stopped on the ground.
Engine vibration might be causal to, caused by, or entirely independent of whatever could also cause a fuel leak. Vibration is often dealt with by operating the engine at a reduced thrust setting, or even shutting the engine down. Troubleshooting often takes more time than you might imagine, and any decisions taken regarding the continuence of the flight or a diversion, will often only occur after that process is complete.
For a heavy aircraft on a long scheduled flight, it is very likely that the maximum regulated landing weight would be exceeded for a considerable part of the initial journey. If a decision to divert is made, then the weight reduction is achieved by opening "dump" de-fuelling valves, that allow a limited amount of the fuel to vapourize at a controlled rate from pipes at the back of the wing. This process can also take a considerable time, and this time management will also be instrumental in deciding on the most suitable diversion airport.
The explanation from the Captain, and subsequently from the airline are both likely to be broadly correct, but neither would be likely to involve very much elaborate detail simply because it rarely achieves the required levels of accuracy and technical comprehension suitable for the general recipients. What might be entirely routine and understandable to a pilot or engineeer, might be truly terrifying to a nervous flyer or the majority of passengers with less specific technical knowledge. You would always strive to be honest, but also ensure information is reassuring (as indeed it should be) and open to the absolute minimum amount of potential misunderstanding.
Similarly, Public relations departments will also have a limited amount of specific technical detail for general information, but would usually attempt to be as honest as possible with the tools at their disposal.
Specific technical details are (depending on the seriousness of an event) often available from public data sources when an investigation has been carried out in accordance with mandatory reporting events.