First of all, please note that when we assign probabilities to a physical events, we are making a mathematical model. Some mathematical models seem to fit reality well. Some mathematical models, which may still be useful, fit not so well. But whether we have good fit or not, we should remember that the mathematical model is not the same thing as the reality.
Let's produce a probabilisitc model of our coin tossing. If there is good reason to believe, because of symmetry, that the two sides are equally likely, then in our model of the situation, we let the probability the coin lands showing $a$ be $p$. Then the probability that it shows $b$ is also $p$. We have $p+p=1$, and therefore $p=1/2$ ($50\%$).
However, that is not the whole story! "Random" in ordinary speech sometimes carries the connotation of "equally likely." But often it doesn't. For example, when you throw two fair dice, the event "sum of the pips is $7$" is a random event. for emphasis, you could say it is absolutely random. However, a sum of $7$ and a sum other than $7$ are definitely not equally likely in the model that experience shows best fits the actual tossing of dice.
When the term "random" is used carefully in a mathematical discussion of probability, it does not carry the additional meaning of "equally likely." If we mean equally likely, we say so.