"This Saturday" surely does mean the closest Saturday in the future. But "next Saturday" has become ambiguous. I've been asked more than once when I used the expression whether it's meant to be the Saturday in this week, or in the week after. (I would usually mean the week after, but if today is Sunday, I might use "next Saturday" to mean the Saturday six days from now. So I find myself being inconsistent.)
Even more subject to confusion is "the Saturday after next", which I'd assume means "the second Saturday from today" -- but I'd want additional confirmation, since not everyone seems to make the same assumption.
Regarding "this Monday" and "next Monday" (or indeed any day of the week), the interpretation does seem to differ depending on what day it is today. If you're making an appointment, I'd suggest confirming the date.
Edit: Inspired by comments, the closest next Saturday can also be identified as "this coming Saturday", and the next following Saturday, as "Saturday week" or (as I learned it) "Saturday a week". As the comments indicate, accurate identification still depends on a (possibly explicit) agreement between speaker and listener.
(Bias: native speaker, of an "older generation".)
The article 'a' or 'the' makes the noun a count noun, which is used to differentiate discrete examples of the noun (which would make them countable) from uncountable uses such as indicating the concept which the noun names.
Examining the prepositional phrases:
In is used here to indicate a state of being by way of being inside a (mass noun used to indicate a) concept. Similar uses of in include in mourning, in thought, and others.
Thus, in this instance in a training is both awkward and incorrect because it conveys that she is inside a training. You might ask yourself, what is one training, and how does it contain a person?
At, on the other hand, is a bit more complex. At practice is common usage, but is used to indicate the physical place at which a person is located. She's at a practice is correct, although sounds awkward because it's unusual not to have a specific, single practice in mind when you tell someone her location. "She's at the practice" sounds less awkward and conveys the same thing. If, however, there are 4 practices being conducted at different places and you don't know which one she's at, she's at a practice is correct to use.
Now, why 'she's at practice' is common usage is a difficult concept. At can be used with a concept to convey a state of being, such as with "at work" or "at play". Here, however, it is meant to convey physical location. One of the things English tends to do is construct nominal phrases- phrases that collectively stand in place of a single noun. It's likely a shortening of something like "at swim practice" where swim practice is a concept with a discrete location and time that the speaker assumes the listener understands through shared context. Not correct usage, but actual usage.
Best Answer
Your use of them is correct.
Local dialects aside, "next Friday" is prospective (looking forward). It's what I would use right now to talk about the Friday a few days from now. By contrast, "the next Friday" is retrospective (looking back). You would use it in past tense, to refer to a Friday that is already in the past.
A clearer example:
Referring to the Friday after next Friday is largely a matter of dialect.
"The Friday after next" is the most common, and is prospective (like "next Friday"). The retrospective variation is technically "the Friday after the next", but I wouldn't use that; it sounds very stilted.
An alternative is "Friday week".