They are not a pair of opposites, as the names suggest, but two terms from different traditions that partly overlap. Hence the confusion.
What each one means
Divergence — a mismatch: price and the indicator move in different directions across neighbouring extremes. The term is settled and means roughly the same thing to everyone.
Convergence — a coming together. And here there is no agreement. Some sources use it for bullish divergence at lows: price falls, the indicator rises, the lines converge. Others use it for price and indicator being in agreement, that is, for the absence of divergence. Those are opposite meanings.
How that happened
Through geometry. Draw lines between two price extremes and between two indicator values, and at lows under a bullish mismatch the lines visually converge, while at highs under a bearish one they spread apart.
That produced the split "convergence at lows, divergence at highs", which other authors never adopted: they call both cases divergence and add "bullish" or "bearish".
Which term to use
Preferably the one that cannot be read two ways. "Bullish divergence" and "bearish divergence" read the same to everyone and need no clarification of what you meant.
The word "convergence" in this context almost always needs explaining, and a term that needs explaining is doing its job badly.
How to tell what an author meant
From the description, not the name. It is enough to ask what price and the indicator are each doing:
- price makes a lower low, the indicator a higher low → a bullish mismatch, whatever it is called;
- price and the indicator move in agreement → there is no mismatch at all.
The second case arguably deserves no name of its own: agreement is simply the absence of an event.
A common mistake
Building conclusions on the pair "divergence versus convergence" as though they were symmetric signals. There is no symmetry: one term describes an event, while the other means a subtype of it in half the sources and its absence in the other half.