TL;DR:
In a couple of weeks it might be possible to say precisely (within a second, perhaps a fraction of a second) when the summer solstice did occur. But until then, sub-minute estimates should be treated as fraudulent.
The reason you are seeing different times for when the summer solstice will occur is that different websites use different models of the orbits of the bodies that comprise the solar system and different models of the Earth's orientation in space.
It's very important to remember that "all models are wrong, but some are useful".
There are several issues with finding the exact time of when solstices occur. One is that the solstices are extrema of the Sun's declination. Without a model of derivatives, finding extrema (e.g., the solstices) mathematically is a much harder problem than is finding zero crossings (e.g., the equinoxes).
More importantly, a very precise prediction would require multiple very precise models. While many models of the orbits of the bodies in the solar system are very, very good, none are good enough to get the timing of the solstices down to the millisecond. Even more importantly, the tilt of the Earth's orientation in space with respect to its orbit about the Sun needs to be modeled. The best model, the IAU 2006/2000A precession-nutation model has over a thousand terms (1365 terms, to be precise).
And even then, "all models are wrong." The people at the International Earth Rotation and Reference Systems Service (IERS) know this very well. The IERS is the organization that is responsible for modeling time as measured by the Earth's rotation and for modeling how the Earth is oriented in space. There are some aspects of the Earth's orientation in space that are not quite predictable. The US Naval Observatory, one of the key members of the IERS, publishes daily updates of observed deviations from model predictions. It take a couple of weeks to fully digest measurements made by astronomers worldwide.
