Can one capture the properties of Turing machine using only function definitions?

There are more partial functions from $\mathbb{L}$ to $\mathbb{L}$ than there are Turing machines in the classical sense of the term. Specifically, there are at least $2^{\aleph_0}$ partial functions from $\mathbb{L}$ to $\mathbb{L}$ (this counts the number of mappings from binary sequences to the singleton sequences $\mathtt{0}$ and $\mathtt{1}$), but there are only $\aleph_0$ possible TMs (each TM can be written as a binary sequence).

Since your definition includes at least $2^{\aleph_0}$ items but there are only $\aleph_0$ Turing machines, your definition can't be capturing all and only the Turing machines.

An example of something not captured: pick any non-recursively-enumerable language (say, the language of all TMs that don't accept their own encodings). Then you can define a (total) function $f : \mathbb{L} \to \mathbb{L}$ as

$$f(d) = \begin{cases} \mathbb{0} & \text{if }d \text{ is the encoding of a TM that does not accept itself } \\ \mathbb{1} & \text{otherwise} \end{cases}$$

No TM exists that (semi)-decides this set, but by your definition this object $f$ would be a Turing machine.

But more generally to your original question - can you capture Turing machines just using functions? - the answer is yes. The $\mu$-recursive functions are a family of recursively-defined (partial) functions that compute exactly the same set as Turing machines. For a slightly more general notion of what a "function" is, you could look to the $\lambda$-calculus, which similarly captures the expressive power of Turing machines using a system based purely on recursive application of functions.