Oho, fighting fire with fire, eh?
You mean i defend from your 'bad playing' in sayin' that you have "bad playing" ?

I see few other options to building a library for culture early game:
Ok, build a library ... But to chop the only trees for that ?
=> Is that a good building to make in the early game?
There are much more interesting buildings to make firstly ...
1) I need the fat-cross
What is it ?
2) Library is more useful than Obelisks, which I very rarely use. Obelisks become obselete too quickly with my tech choices, and Libraries will always pay for themselves in coastal cities.
Library gives 2 culture whereas obelisks gives 1.
The interest of these buildings is to reach 10 culture enough fastly (to expand), the rest isn't important, so having obelisks is quite enough, library will be useful for science after.
Obsolete is a problem, but you won't need more culture, often.
(If the obelisk is obsolete, do you lose his +1 culture income ? or you keep it but only loosing the ability to build others ?)
Note that obelisks cost 30 and library 90 ...
Knowing that obelisks are better to make a faster culture expand by his little cost, they are the best choice, for early culture expand.
Library in the early game is useless, it costs too much for its culture, and the science bonus isn't worthwile when you don't produce enough science in the cities, you don't need the library.
Why the library are a good choice for coastal cities ?
The libraries make culture and science ...
You talk about ocean's gold? The libraries will give a very bad income for the 2 gold you gain, in the early game.
3) I had no state religion, and had yet to adopt the Mongolian religion
For culture, obelisks are good if you haven't any state religion

You don't need more 1 culture expand...
Obelisks are quite enough

Regardless, if trees are going to be chopped, what makes a Library a bad choice for a city of middling size that has no hope of ever going for a Wonder
You must not chop trees, they are good unchopped, +1 prod and more with high technologies is better than having 50 hammers for only 1 thing.
Particularly when you have only 1 tree, you won't chop it, if you are smart

But if you have to chop these trees, that's to make a high cost military unit, or something that has a REAL use, instead of an useless library.
A risk that pays off is good playing.
A risk that ends in disaster is bad playing.
I think that in your case, you had a disaster, so that's bad playing

Just kidding
??
Granted, it would be less offensive to say 'mistake' rather than 'bad playing'. You all have my sincerest apologies for my lack of tact.
Yes, and even if this is a mistake, this mistake is a really good exemple, to make creators & developpers change this part of the game.
a weak defender in an important city
If the city isn't important, that's normal that you can abandon it for some turns waiting for reinforcements that will defend the city in their place, but if the city is important, you shouldn't give it up no matter/whatever the reasons, or only you are SURE that you won't lose it.
But that's strange that the AI sent units from his last city, to capture your city, whereas he should defend his city ...
If one is forced to resort to such drastic measures, then that person has failed to properly plan military needs.
You mean that someone who send defenders and who abandon cities to put these soldiers on the frond has failed?
I don't think so.
If you want to make a big assault, with the maximum troops, using your defenders can be a good help for your attackers.
This isn't bad.
But you must pay a big attention to your cities, replacing the lost defenses fastly, and you can't afford this "defenders => attackers" if your cities are in danger.