Is the battery glued in? MacBook Pro (Retina, 15-inch, Early 2013)

Probably, but it is replaceable. See MacBook Pro 15" with Retina (Mid 2012 - Early 2013) Model ID: MacBookPro10,1. OWC provides instructions and tools needed to replace the battery. They sell Apple batteries, if they are available.

It is still possible to have an Apple Authorized Service Provider to perform an official Apple battery replacement even though the laptop is considered "Vintage" as long as Apple still has the part in stock. I highly recommend an official Apple battery since the quality of third party replacements is usually quite poor even when purchased from a reputable vendor such as OWC.

That is good to know. I did not know OWC sold Apple branded batteries. Are you able to specify which one you get and are the Apple branded ones regularly in stock?

Over the years I've gotten a lot of defective Lithium-ion batteries from OWC (the OWC store brand).

I think it may be dependent on whatever inventory OWC has obtained. Some shops like theirs buy large lots of unused Apple parts that Apple no longer makes for obsolete hardware. Same with other companies, like NuPower, that no longer carry inventories of batteries for obsolete hardware. Operations, like OWC, pick these up in the gray markets where they are sold by auction. Of the four batteries I bought from OWC, three went into two 2009 MacBook Pros 15'' models and one 17'' 2009 MBP. The fourth replaced the battery in a 2015 12'' MacBook. This last one was Apple-branded. I think the one for the 17'' MBP also was Apple-branded. YMMV, of course.

I'm pretty good about shopping around for good deals on components I need to replace in any of my Macs. You never know what might fail. I don't do that much any more because my hands get shaky when I need to work in close quarters or with small screws and parts. I have a lot of problems working on hardware, now, that I did not have many years ago in my 50s and 60s. But old age can start to eat away at you once you hit the 70s.

I must, respectfully, disagree in part. I have purchased several batteries from OWC. Two were Apple branded. The other two were from NuPower, also branded. All were new replacements, not refurbs.