Schottky Rectifier vs Schottky Barrier

Here is a Diode Question:

What is the difference between a Schottky Rectifier and a Schottky
Barrier diode?

Both have relativly low forward voltages, but when would I use one vs
the other?

The reason I'm askingis because a Schottky Rectifier was designed into
a product and used as a blocking diode to prevent a Li-Ion Coincell
from charging. But it has since been found that the reverse leakage of
this Schottky Rectifier diode was high enough that the Li-Ion battery
was charging. I replaced the Schottky Rectifier diode with a Schottky
Barrier BAT54 diode and the problem is gone.

In general do Schottky Rectifier diodes have higher reaverse leakage
current than Schottky Barrier diodes?