Learn English – When did saying “I’m offended” or “I find that offensive” become a common phrase in English

The OED recognises both uses of the verb 'deliver', under their meaning 3.

In both cases, and fundamental to the meaning of 'deliver', is the idea of freeing and releasing. Note that they refer to the second (or b) as the passive.

a. To disburden (a woman) of the fœtus, to bring to childbirth; in pass., to give birth to a child or offspring. Rarely said of beasts. (The active is late and chiefly in obstetrical use.) †b.pass. Of the offspring: To be brought forth (lit. and fig.). Obs.

This has been an excellent question.

It is certainly still current usage, in my experience in Britain, and at least in print, to speak of the mother as the one who has been 'delivered'. ('She was delivered of a nine-pound baby son'.) But you also see 'the mid-wife assisted in the delivery of a baby girl'. Indeed it not uncommon I believe to speak of the doctor or mid-wife as the one who delivers. It now seems clear that the doctor can both deliver a mother (from the rigours of childbirth), and a child (into the world)

In answer to the question as to when the mother is first spoken of as the 'deliverer' as opposed to the 'delivered', the first OED reference is from 1581.

1581 G. Pettie tr. S. Guazzo Ciuile Conuersat. (1586) i. 12 All beastes so soone as they are delivered from their dam get upon their feete.

There are also a couple of references from Shakespeare circa 1609.

Edit 4/July/2020 Reviewing this matter again after six years I believe @Peter Shor summarises the position succinctly in his comment thus:

If you search Google books for "delivered a child", in the 19th century, it's always a doctor or a midwife delivering the child. In the early 20th century, you see that some mothers have started delivering their own children. But in the 19th century, it's doctors either delivering a child of a woman, or delivering a woman of a child.

I now believe that @Blckknght made a pertinent contribution when he said:

This is only a guess, but I wonder if mother-as-deliverer arose due to drift in the meaning of "deliver". I don't think any modern speaker would use "deliver" to mean "disburden" or "liberate" these days, as the "carry" and "hand over" usages have taken over. If "deliver a baby" was already a phrase in the collective vocabulary, it might make sense for the exact sense of who is doing the delivering to change along with the meaning of the word "deliver".

There has clearly, across the centuries been a drift in the meaning of "deliver". Though to me it seems that the two meanings of deliver - a) to liberate someone of something and b) to hand over something to someone, have co-existed across time. There is an OED example of b as early as the thirteenth century.

Thanks to Peter's succinct summary therefore:

a)In the 19th century, it's always a doctor or a midwife delivering the child. b)In the early 20th century, you see that some mothers have started delivering their own children. c) But in the 19th century, it's doctors either delivering a child of a woman, or delivering a woman of a child.

Added to this is the additional curiosity that so far as the animal kingdom is concerned, mothers were apparently delivering offspring as long ago as 1581.

1581 G. Pettie tr. S. Guazzo Ciuile Conuersat. (1586) i. 12 All beastes so soone as they are delivered from their dam get upon their feete.

OED has a very early citation:

c1394 P. Pt. Crede 375 Þey ben digne as dich water þat dogges in bayteþ.

In this case, digne doesn't mean dull, it's related to dignity and OED has it defined as "Having a great opinion of one's own worth; proud, haughty, disdainful; (cf. ‘stinking with pride’)".

Ditchwater is generally muddy and not clear: it's dull. And it can be smelly. The translation appears to be "They are as smelly" [or "Their pride stinks"] "as ditchwater that dogs have drunk from."

Google Books has "dull as dishwater" appearing in The Amaranth published in Boston in 1854.