Choice between dummy variables and Likert scale in Linear Regression

In your question you talk about coding so I will address primarily the question of coding.

You should definitely code it using suggested

0 for a, 1 for b, 2 for c and 3 for d

regardless of whether you plan to use it as ordinal variable or categorical variable.

Let me explain, nowadays virtually any language or program will have option to easily turn ordinal, or categorical variables into dummies with virtually zero effort.

In R you can simply use is.factor() function to turn any ordinal variable into separate dummies so you could use:

lm( dep_var ~ ind_var + is.factor(variable), data = XY)

In Stata you can use i.variable to turn any ordinal or categorical variable into dummies, again in Stata you would run something like:

reg depvar independentvar i.variable

Next, coding it as separate dummies is more time consuming (even if just marginally it takes more line of code) than just coding it into one variable.

Consequently, no matter whether in actual model you want to include it as oridinal variable or set of dummies, in most standard languages and programs it will be more efficient (i.e. take less time to code) if you just code it on Likert scale.

As for whether what you should do from scientific standpoint that depends on details of your research. I can imagine you would be being able to defend both including the variable as independent ordinal scale or as a separate dummies. Both of those would work in principle. My sense is that you would likely want to include it as dummies since that will have better interpretation, but if you have little data and too many regressors then it might be worth while to just have it as ordinal variable.