Are you tired or sleepy?

For Wiktionary, which in this case borrows the definitions from the Oxford Dictionary, “tired” and “sleepy” are synonyms. However, in other dictionaries and sites, the definition of “tired” is slightly different:

1. From Babylon:
"fatigued, dead-tired, exhausted (Informal)"

2. From Wikipedia, not Wiktionary, where it is a synonym of fatigue:
"It is ubiquitous in everyday life, but usually becomes particularly
noticeable during heavy exercise. Mental fatigue, on the other hand,
rather manifests in somnolence."

3. From Wordnet:
"depleted of strength or energy;"

In these cases, sleepiness might be just one manifestation of being tired, but someone can be sleepy without being tired (when taking a pill, for example). Therefore, in some circumstances there can be a causal relationship but these are independent words in my opinion.

I know this is a rather silly discussion but I started it with a friend the other day and now I want to settle it down. I even opened a thread on Wiktionary, proposing to change their first definition of the word “tired”.

Of course I’m ignoring the other uses of the word which don’t apply here, for example:
1. Fed up, annoyed, irritated, sick of: “I'm tired of this”
2. Overused, cliché: “a tired song”