First, here are the stories in past and present:
Imagine there is an appointment at a caffe. I am 10 minutes late arriving at the caffe. My friend is already in there. I say: "I'm so sorry! I am late". My friend says: "No problem, I just arrived".
There was an appointment at a caffe. I was 10 minutes late arriving at the caffe, and my friend was already in there. I said: "I'm so sorry! I am late". My friend said: "No problem, I just arrived".
Which tense is better?
If it's a hypothetical, imaginary, and very brief story which is just to illustrate the use of a word, then the present tense seems like a pretty good choice.
If it's a longer imaginary story, which is meant to tell a story, then the past tense is preferable.
If it's a true story that actually happened, you have two choices.
- the past tense (obviously), or
- the historical present or historic present, also called dramatic present or narrative present, which is the employment of the present tense when narrating past events.
In summary, for exactly the type of scenario you are asking about, the present tense seems like a good idea.