Your understanding is correct, and you are right that in the hospital, you would likely have had a cesarean or a instrumental vaginal delivery. Here is a systematic review on the subject. (A systematic review is a structured analysis of all reresearch on a particular topic using predetermined criteria.) Short version: as long as mother and baby are tolerating labor, it can continue. At some point, though, the judgment call must be made that this baby is not coming out by itself, and the decision will be made to deliver the baby either via cesarean surgery or instrumental vaginal delivery. In hospitals, that decision is likely to be made a lot sooner than it needs to be. If you think the second midwife is open to information that contradicts what she currently believes, you might send her the citation and study summary (aka "abstract" in academic lingo).
-- Henci
P.S. It is not surprising that maternal injury goes up with length of second stage because instrumental vaginal delivery, especially in combination with episiotomy, increases the risk of deep tears, although if it is feasible to deliver the baby vaginally, it is a better option than running the risks of cesarean surgery.
Altman MR, Lydon-Rochelle MT. Prolonged second stage of labor and risk of adverse maternal and perinatal outcomes: a systematic review. Birth 2006;Dec;33(4):315-22.
BACKGROUND: Safe and effective management of the second stage of labor presents a clinical challenge for laboring women and practitioners of obstetric care. This systematic review was conducted to evaluate evidence for the influence of prolonged second stage of labor on the risk of selected adverse maternal and neonatal outcomes. METHODS: Articles were searched using PubMed, Cochrane Library, and CINAHL from 1980 until 2005. Studies were included according to 3 criteria: if they reported duration of the second stage of labor, if they reported maternal and/or neonatal outcomes in relation to prolonged second stage, and if they reported original research. RESULTS: Our systematic review found evidence of a strong association between prolonged second stage and operative delivery. Although significant associations with maternal outcomes such as postpartum hemorrhage, infection, and severe obstetric lacerations were reported, inherent limitations in methodology were evident in the studies. Recurrent limitations included oversimplified categorization of second stage, inconsistency in study population characteristics, and lack of control of confounding factors. No associations between prolonged second stage and adverse neonatal outcomes were reported. CONCLUSIONS: The primary findings of our review indicated that most of the studies are flawed and do not answer the important questions for maternity caregivers to safely manage prolonged second stage. Meanwhile, approaches for promoting a normal second stage of labor are available to caregivers, such as maternal positioning and pain relief measures and also promoting effective pushing technique.