Is a tweakable block cipher still considered deterministic in nature?
When talking about encryption with block ciphers, you should distinguish block cipher itself from the mode of operation that employs the block cipher. 

Block cipher takes an input $P$ of fixed length $n$ and transforms it under key $K$ (and possibly under other parameters such as tweak $T$) to output $C$ of the same length $n$:
$$
E:\; P \xrightarrow{K(,T)} C.
$$
It is always deterministic in the sense that $C$ is entirely determined by the listed parameters.

A mode of operation $\Pi$ is used to turn a fixed-length block cipher into a variable-length encryption scheme. To achieve semantic security, modes of operation typically uses deterministic nonces or random initialization vectors. In the latter case the resulting schemes are called probabilistic. 

The use of tweaks is somewhat orthogonal to this classification. Both deterministic and probabilistic modes may use tweaks to further parametrize the encryption (e.g., in disk encryption).