The original Poster is quite right. DO in these examples is lexical or 'main verb' DO, not the auxiliary verb DO:
I do my homework.
You do the laundry.
We do the washing up.
They do yoga.
He/she does the cleaning.
We can show this by doing some tests. To make negatives in English we put not or n't after the auxiliary verb. But this will not work with these sentences:
- *I don't my homework. (ungrammatical)
- *You don't the laundry. (ungrammatical)
- *We do not the washing up. (ungrammatical)
- *They don't yoga. (ungrammatical)
- *She doesn't the cleaning. (ungrammatical)
Also we cannot make questions here by inverting the subject and DO. If DO was an auxiliary here then this would be possible but it itn't:
- *Do I my homework? (ungrammatical)
- *Do you the laundry? (ungrammatical)
- *Do you the washing up? (ungrammatical)
- *Do they yoga? (ungrammatical)
- *Does she the cleaning? (ungrammatical)
If we want to make grammatical questions and negatives here we need to use the auxiliary verb DO as well as the lexical verb DO:
- I don't do my homework.
- You don't do the laundry.
- We do not do the washing up.
- They don't do yoga.
- She doesn't do the cleaning.
- Do I do my homework?
- Do you do the laundry?
- Do you do the washing up?
- Do they do yoga?
- Does she do the cleaning?
Also if we want to give emphatic versions of the sentence we need to add an auxiliary DO to the sentences:
- I do do my homework!
- You do do the laundry!
- We do do the washing up!
- They do do yoga!
- She does do the cleaning!
All of this data shows that the Original Poster's examples use main verb DO and not the auxiliary verb.
The site that the OP links to is published by a language school company. Never go to one of these schools!
Note For some people yoga seems to be a verb. In this case, They do yoga would be an emphatic version of they yoga! Don't copy this in writing because it isn't standard (yet!)