Hi, I’m wondering how people handle online games where not everyone has “the book”. Some games have an SRD, or have everything on a shareable character sheet, but I’m wondering what you do for games that don’t.
Do you write up your own summary? Can you share the book without allowing it to be downloaded by the third party?
I prefer to write up the core rules onto tabs of a character keeper. I’m uncomfortable about sharing an entire pdf and to be honest I’m not certain players need it. For quite gritty games where you feel the work to create a summary is too much I’d think about sharing copies of the reference sheets most games worth their salt come with.
PS: Welcome to the community
I just share the rulebook with the other players at the table, if they ask for it or we feel it could be useful for the game.
No fuss 
The only occasion in which I would write a summary instead, is if the original rulebook is so badly written or organised, that a summary would make the game experience better than relying on the original text.
(uncommon, but it has been known to happen)
In the privacy of your home table, you do you. In my opinion 
For online play, I generally have a cheat sheet (a lot of games have their own, some character keepers have them, or I’ll make one) which I keep in a Google Drive folder for the campaign.
During Session Zero I’ll generally do ‘the teach’ and run through the rules using Jamboard or pages on Roll20, to get some of the core ideas across.
To me, it depends on the game.
If a game really requires everyone to have a copy - I’m thinking of Firebrands games, for example - I’ll share the pdf with the players and explicitly say “you’ll need this text to play, but if you enjoy the game, I encourage you to support the designer by buying a copy”, with a link to the page where they can do so. I consider this a normal and as-intended use of the game text, and don’t have any reservations about sharing in this way.
If there are specific parts of the text players need, but not the whole thing, I’ll print those parts to pdf and share them in a google drive. For example, I often do this for skill descriptions in games like Fate or Gumshoe.
If the players don’t need the text at all, then I’ll just aim to explain things verbally. I play a lot of pbta games were everything the players need is on their playbook or the reference sheet, for example.
I’m with Sandy on this one. I think sharing a PDF for the purpose of conducting a game is fine. I also lend my PDFs to other players, e.g. to create a character. I expect them to delete the copy when there is no longer any need to keep it or buy their own copy.