Insist on doing something (v.intr.) = To be resolute or firm in a demand or course: He insisted on paying for his share of the expenses.
Insist that (v.tr.) = To assert or demand (something) firmly or persistently; and often that something is true, especially when other people think it may not be true:
He insisted that he’d pay for his share of the expenses.
He merely insisted on paying for the dinner but didn’t take his wallet out.
Notice that we never say that someone ‘insists to do’ something.
He merely insisted to pay for the dinner but didn’t take his wallet out. ✗
Jane insisted on being posed for photograph next to the Statue of Liberty.
It was cold and wet outside so he insisted on staying at home.
Hilda had a painful and upsetting divorce – She doesn’t want to talk about it. I don’t understand why you insist on talking about it.
It has rained all day. I don’t know why you insist on watering the flowers.
The construction insist + infinitive is believed to be nonstandard which deviate from a commonly accepted language norm, and many native speakers would probably say it is not conforming to grammatical rules. The standard format is insist on + present participle:
If you insist to go out in this rain, you must take the umbrella. ✗
If you insist on going out in this rain, you must take the umbrella. √
In constructions such as “Jane insisted [that] Linda stayed the night“, most discreet speakers would use the subjunctive form “Jane insisted [that] Linda stay the night.” Here the subjunctive doesn’t “conjugate” (for different tenses, or Ist/ IInd / IIIrd person subject, etc.) in the same way as the infinitive.
My wife insisted that I stay with the kids for a couple of weeks before going back to work at Stanford.
This kind of construction is called ‘mandative subjunctive’ and is used to express circumstances that are desired, demanded, etc.
Many credible writers on ‘subjunctive’ point out that there is a possible confusion between the ‘indicative’ and ‘subjunctive’ mood in sentences with propose, insist and suggest:
She insists that he is at home for Christmas (indicative, a forceful assertion of the fact that he is at home).
She insists that he be at home for Christmas (subjunctive, a demand that the condition of his being at home be fulfilled).
Notice that do/does are not used in negative subjunctive sentences:
The doctor insisted that he doesn’t walk briskly until he has his stitches taken out. ✗
The doctor insisted that he not walk briskly until he has his stitches taken out. √
Insist on my/me (doing) = both are correct.
She insisted on my telling a joke. (preferred choice in writing; telling a joke is a gerund, a noun, so it takes the possessive my).
She insisted on me telling a joke. (informal; in spoken English)
Hence, “She insisted on my telling a joke ” and “She insisted on me telling a joke” are on an equal footing.