Sure. I think the information you have applies only to options on publicly-traded stock because the value of stock that is not publicly traded is sufficiently hard to determine to prevent it from becoming a routine or traditional part of an employee's compensation package. One way to value a non-traded stock is to use the valuation used in investments, but there are not enough investment events for the phrase "the value of the stock the day you start" to make sense -- especially when you take into account that the terms of many investments are never published (with the result that the employee would need to take the startup's word on the terms).
That's how I conclude that the information you give in great grandparent does not apply in the current situation :)
From my experience, stock options are priced based on start date(correct me if this is not standard). While the price changes less frequently for private stocks compared to public stocks, they still change (Typically through investment events as you mentioned). Thus, if this particular startup raised a seed round in the coming months, then the value of the stock options would not rise as op mentioned - "The option % they've offered me would only rise in value" - since the pricing of the stock options would be different.
