Shall a $1^{\rm st}$ order IIR filter include $x[n-1]$ term?
A filter described by the difference equation

$$y[n]=bx[n]+ay[n-1]\tag{1}$$

is just a special case of a first-order filter. Its transfer function is given by

$$H(z)=\frac{b}{1-az^{-1}}=\frac{bz}{z-a}\tag{2}$$

from which you can see that it has one pole (at $z=a$) and one zero (at $z=0$). A more general first-order filter has a zero that is not necessarily at the origin of the $z$-plane:

$$H(z)=\frac{b_0z+b_1}{z-a}\tag{3}$$

Now the zero is at $z=-b_1/b_0$.
This corresponds to the more general first-order difference equation

$$y[n]=b_0x[n]+b_1x[n-1]+ay[n-1]\tag{4}$$

You can see that $(1)$ and $(2)$ are just special cases of $(4)$ and $(3)$ with $b_1=0$.
So both filters you mentioned in your question are first-order filters, but the first one is a special case because its zero is at $z=0$.