I'm having some confusion on the correct implementation of the Backward Euler method. Assume we have a general 1st order ODE given in the form:
$$ \frac{dy(t)}{dt} = f(y(t),t) $$
Typically I recall it is implicit, meaning we expect a given step to be calculated from: $$y_{n+1} = y_n + f(y_{n+1},t_{n+1}) \, \delta y$$
However, in implementing Forward Euler and Backward Euler side-by-side, it seems like I can, given initial conditions $y_0$ (and $y_1$ for BE), choose to approximate the derivative with either a forward step or a backward step. write these as:
$$ FE: \qquad y_{n+1} = y_{n} + \left[ \frac{y(t+\delta t) \,-\,y(t)}{\delta t}\right] \delta t$$
$$ BE: \qquad y_{n+1} = y_{n} + \left[ \frac{y(t) \,-\,y(t-\delta t)}{\delta t}\right] \delta t$$
Where we Taylor expand the terms in the brackets to first order. This seems to allow us to have a simple explicit scheme using either a forward or backward difference approximation, but seems not to line up with the expected format.
My question then is - is this Backward Euler? If not, what have I done and what is wrong with it? (It seems to work well enough). And how would I change this to get to an implicit scheme (which I believe would need the solution of a fixed-point problem at every iteration step).
Perhaps I am getting confused with applying the concept to different use-cases, if Euler methods are used differently when not solving 1st order ODEs. (Indeed - are they used at all for other purposes or are they purely for integrating ODEs?)