Cloud Computing is the delivery of computing as a service rather than
as a product, whereby shared resources, software and
information is provided to computers and other devices such as
utility (such as the electricity grid) through a network (usually the
Internet) Clouds can be classified as public, private or
hybrids.
So CloudComputing is not the product delivered but the availability of services over a network.
These service models can be (according to the same Wikipedia article) :
Infrastructure as a service (Infrastructure as a service (IaaS))
Iaas providers offer computers (physical or virtual machines) and other resources, load balance, fail over, ability to scale services more or less depending on customer demand or need, storage with distributed backups, VLANs, firewalls.
Platform as a service (Platform as a service (PaaS))
In the PaaS model, cloud providers deliver a computing platform that typically includes the OS and an environment for running a programming language, databases, and Web servers. With this model, developers can deliver their applications without having to worry about the layers of hardware and software and to focus only on developing their products.
As an example we have Microsoft Azure, Google App Engine, which scale processing and storage without requiring manual intervention of the developer.
Software as a Service (Software as a service (SaaS))
In this business model users have access to applications and their databases. Cloud providers provide the IaaS and PaaS that run the application. SaaS are generally considered "on-demand software" because they are paid per use.
In the SaaS model providers install and manage applications and software users access their services and pay by period or by consumption.
Thus:
One of the first applications that came out completely online is the
emails, like today we have Gmail, Outlook, etc., fully accessible
via browser. I've never seen anyone call this a "cloud", but have they not
would be the case, since this works without having to install
anything on the machine?
Install is not the point, email can be considered a SaaS and yes, it would be a Cloud because it is an application sold for you to use under the responsibility of another company
If a business owner resolves to release external access to the ERP in their
server, so that some people can access from their homes, this
already characterized as a cloud? If not, what would it take for this?
be a "cloud", put this on a third-party server?
No, it would be naked if the company's business was the ERP and it would sell access to it to other companies.
Could a simple website, such as a blog, be considered a mini-application in the clouds?
No, because it does not provide any service to your users