Comparing ML models to baselines
This depends on what you want to show.
When working with metrics you shouldn't just take the value as is, but see what each metric are telling you. baseline_1 isn't better/worse than baseline_0 because it has a higher/lower value in metric X. Both baselines give an interesting perspective on a given dataset and if unsure I'd suggest keeping both.
A couple of notes:

when saying baseline, I will refer to the two baseline strategies that you said in your post
I will use the accuracy metric for examples but what I'm saying is true for any metric.

Why use baselines?
People usually tend to see accuracy (or other measures) as absolute values. E.g. accuracy=0.9? &quot;very good&quot;, accuracy=0.3? &quot;very bad&quot;. This isn't true however, as metrics are influenced by the number of classes and the proportion of samples between them.
However an accuracy of 0.3 in a classification task with 1000 classes is arguably much harder to achieve than an accuracy of 0.9 on a binary classification task (assuming class balance in both cases).
Here is were baselines come in. They can show how much better a model is than a dump classification strategy.
How baselines help?
Baselines help by putting a lower bound on your metrics. For example an accuracy of 0.55 on a binary classification task is slightly better than random, but the same accuracy on a 10-class setting is much better. Baselines help quantify that and tell you how much better you are than predicting random  or the most common values.
What effect do baselines have?
Now on to why keep both baselines:

The first baseline (i.e. random) helps show how metrics can be influenced by the number of classes has on the dataset.
The second baseline (i.e. most common) helps show how metrics can be influenced by class imbalance.

How baselines actually help?
Let's you have two models, one with an accuracy of 0.92 and another with an accuracy of 0.93. How much better is the second model to the first? This depends on the value of your baseline. If you have a baseline accuracy of 0.5 then both models are relatively strong and the difference is not that significant. If you have a baseline of 0.9 then the models aren't as strong and an improvement of that magnitude is more significant.