Question about machine epsilon
There is definitely an inconsistency in what you've written here. My guess is that $t$ doesn't mean what you think it means. You are using it here to describe the size of the significand field, that is, the number of digits after the radix point. (It's not a decimal point in general!)

My guess is that your lecturer is using it to refer to the precision. The precision of the floating point format includes the implicit $1$. That is, your floating point number is:

$d_1.d_2 d_3 d_4 \ldots d_t \times \beta^e$

where $d_1$ is always $1$. If you interpret $t$ that way, the two epsilons make more sense.

This is the way that the IEEE-754 standard defines these constants. For example, a binary32 (formerly called "single precision") floating point number has a precision of 24 bits, in that the mantissa is 24 bits. However, the significand field is only 23 bits, because the leading one bit is implicit.