LSTM with input of actual time step
I'm not sure whether I understand the question, but if you want to include var1(t) along with the other lagged features var1(t-1), var2(t-2), etc. you can simply add var1(t) as an additional feature.
This means that your matrix with previous shape

(number of samples, number of timesteps, old number of features)


now becomes

(number of samples, number of timesteps, new number of features = old number of features + 1)


Judging from your code this can be simply done by adding var1(t) as a new column to your train_X matrix, like in the first three columns of your first code snippet.

However I'm not sure why you want to do this. Forecasting at time t usually assumes that, once your model is ready to work on real data, you won't have any data at t, only from the past. If you build your model based on data at time t without being able to retrieve these information once in production, then you've got a problem! Also, you write that var1 and var2 are correlated, hence the LSTM is already picking up this correlation as you are feeding past values of these variables together.