A Dictionary of Physics (Oxford University Press) defines acceleration as:
The rate of increase of speed or velocity
However, from reading many other definitions it seems to me that acceleration in Physics is generally held to refer to the rate of increase of velocity (rather than speed), and that acceleration is generally held to be a vector quantity rather than scalar quantity.
How does this relate to the definition above of acceleration being able to be an increase in speed? What are the contexts in which acceleration is treated as an increase in speed? Do these tend to be more simplistic, less real-world treatments of acceleration or do they belong to another field such as Mathematics where the treatment may be more abstract and direction can be ignored?
I note that the SI unit for acceleration is metre per second squared (m/s2). Is it significant that this unit does not specify direction? Does this allow for accelertion to be a scalar or vector quantity depending on whether or not a direction is specified?