If a language has a regular grammar, is it regular?
A regular language can be defined as &quot;a language with a regular grammar&quot;. So, yes, the existence of a regular grammar for $L$ means that $L$ is regular. [Note 1]
That doesn't mean that every grammar for $L$ is regular. That doesn't matter for this purpose.
It's important to keep the distinction between languages and grammars clear. In general, we say that a language is X if it has at least one grammar which is X (deterministic, regular, context-free, etc.).
Sometimes, the categorization of grammars is finer (makes more distinctions) than the categorization of languages. A grammar can be left regular or right regular, but a language is simply regular: if it has a left regular grammar, it must have a right regular grammar, and vice versa. Similarly, a grammar might be LR(1), LR(2), etc., but this distinction doesn't apply to languages because if there is an LR(k) grammar for a language, there is also an LR(1) grammar.
This distinction is very important when discussing the relationship between different categories. Failing to make the distinction correctly leads to confusion, because the contexts are very different. For example, there is a simple linear-time algorithm which will tell you if a grammar is regular. However, there is no algorithm which can tell you whether the language recognised by an arbitrary context-free grammar is a regular language.

Notes

I believe the original definition of regular languages was based on Kleene's regular expressions. The concept of regular grammars comes from Chomsky's seminal work, in which he called them &quot;Type 3 grammars&quot;. His 1959 paper which introduces this hierarchy of grammars (and, indirectly, of languages) contains the proof that Type 3 languages are the same as what he called &quot;finite state languages&quot; -- languages which can be recognised by a Finite State Automaton.
This equivalence leads some people to define &quot;regular language&quot; as &quot;a language which can be recognised by a FSA&quot;, although that definition fails to justify the use of the word &quot;regular&quot;. (It is the definition found in Michael Sipser's Introduction to the Theory of Computation, and also on Wikipedia, for example.)
It is also easy to show that the set of languages which can be described by a regular expressions (in Kleene's sense) is precisely the same as the set of languages which can be described with a regular grammar.