Is there something called “mean coding” (like dummy coding & effect coding) in regression models?

When we perform a regression analysis with categorical predictors, we can use (1, 0), called "dummy coding". The coefficients in this case represent the deviation of the groups' means from the mean of the reference group. We can also use (1, 0, -1), called "effect coding" the coefficients in this case represent the deviation of the groups' means from the grand mean. But what if we did our analysis by excluding the intercept column from our matrix and without setting a reference group? $$Y_{ij} = \mu_j + \varepsilon_{ij}$$ with $j = 1, \ldots, k$ factors, and $i = 1, \ldots, N$ observations. If $X$ is the design matrix, the model could be: $${\bf Y} = {\bf X}u + e$$ $${\bf Y} = \begin{pmatrix} Y_{11} \\ \vdots \\ Y_{1j} \\ \vdots \\ Y_{Nk} \end{pmatrix}, \qquad {\bf X} = \begin{pmatrix} 1 &0 &\cdots &0 \\ \vdots &\vdots &\cdots &\vdots \\ 1 &0 &\cdots &0 \\ \cdots &\cdots &\cdots &\cdots \\ 0 &\cdots &0 &1 \\ \vdots &\cdots &\vdots &\vdots \\ 0 &\cdots &0 &1 \end{pmatrix}, \qquad \mu = \begin{pmatrix} \mu_1 \\ \vdots \\ \mu_k \end{pmatrix}$$

Then the coefficients would represent the groups' means, is that right? Would that be average or mean coding?

There are a couple of things to be aware of when you use level means coding. First, you must suppress the intercept to avoid having perfect multicollinearity; see: Qualitative variable coding in regression leads to “singularities”). Second, the meaning of the hypothesis tests is different: they are now tests of whether the means differ from $0$, not whether they differ from each other; see: Understanding dummy (manual or automated) variable creation in GLM.