Let $k$ be a commutative ring with unity and $A$ an associative algebra over $k$. The algebra of formal power series $A[[t]]$ with coefficients in $A$ consists of all formal sums $\sum_{n\geq 0}a_n t^n$, where each $a_n$ is in $A$, and becomes an algebra over $k[[t]]$ under the usual definitions of addition, multiplication and scalar multiplication.
In this MathOverflow question, it is proved that for a field $k$ the canonical map $A \otimes_k k[[t]] \to A[[t]]$ is only injective in general.
On the other hand, some papers on deformations of algebras seem to take $A \otimes_k k[[t]]$ as the definition of $A[[t]]$, see for example the page 19 in An introduction to algebraic deformation theory by Thomas F. Fox or the page 135 of Algebraic Deformation Theory by W. Stephen Piper.
I have two questions:
1) Why are they taking $A \otimes_k k[[t]]$ as the definition of $A[[t]]$?
2) What is the most widely used definition of a formal deformation of an associative algebra?