It may be we're using different terms to describe the same thing, or not....
I just get riled when `how to cast on' instructions blithely assure a new knitter that `the longtail always makes the first row, so you start your pattern on the next row'. That just isn't true, and causes a lot of confusion in beginners. A cast on is a cast on, and not a `cast on plus one row'. Everyone should try different methods and see what each type of cast on produces, then decide for themselves whether they should do another row, start in with the pattern, or do it another way. Knitting is very individual and nothing should be so specific that people feel they're doing something wrong.
I can see how that would irritate you. It is really much better to explain that the long-tail cast-on has a knit side and a purl side, and let the knitter decide what to do from there. I suppose the "plus one row" is merely more advanced knitters' shorthand for the effect.
Structurally, however, I must say that at least my long tail cast-on *is* a backwards loop cast-on that has been turned and purled tbl to the end. It's neater and tighter, but the path of the loops is identical. It's like the difference between a ssk and a sl1, k1, psso. Made differently but the end result has the yarn in the same relationship.
In aid of this, I've got a picture of myself forming a stitch in long tail cast on, and also myself forming a backwards loop. Monste Stanley has it labeled the twisted loop method and advocates working through the front of the stitch to give the twist in the foundation row. I prefer working it through the back of the stitch, however, which should clarify why they turn up the same edge if I purl back the first row. The long-tail is still a firmer edge and tends to be neater, of course. The gauge difference between the loops does affect how the edge behaves.
Just to add some more info into the mix, both knitting on and the cable cast-on tend to be inelastic; knitting on more so than the cable CO. Of the two, I prefer the cable CO as I find it looks neater and has a more solid edge. I rarely use it, however, as I find the long tail CO sufficient for most things where I need a stable edge, and it's much less visible. Meanwhile, the alternate cable CO is a variation which is just as stable as the cable CO, but more elastic and well suited to ribbings.
Elasticity and suitability of particular cast-ons matters if you think about things like socks or mittens or hats, neck edges, sleeve cuffs, etc. Sometimes the edge needs to stretch, sometimes it doesn't matter, and sometimes it really needs to be firm and solid.
So, I'd suggest a collection of cast-on methods instead of one favorite. In no particular order: *Long tail - suitable for well-nigh everything. Stable, elastic, neat.
*backwards loop - useful for casting on in the middle of a row without joining extra yarn, and very fast to implement. Very unstable and can be loose and messy, but practice generally improves that. Useful when swatching if you don't want to work a more time consuming or elaborate cast-on.
*cable and alternate cable (which should be very simple to learn since you already know how to knit on) - Also suitable for casting on stitches in the middle of a row, very stable decorative edge. The alternate method is elastic enough to use for sock cuffs, although I wouldn't recommend using the standard method for that purpose.
*the bind-off cast-on which uses a crochet hook - it's a handy way of making the cast-on edge look the same as a standard psso bind-off. Not a particularly "special" or imperative reason, but it's a simple touch which can help make something look much more polished.
Once you're comfortable with these, I'd add in: *one of the provisional methods because casting on provisionally pops up in all sorts of places. *and if you're interested in mittens or socks, the figure-8, which is closely related to one of the provisional methods and is very simple.
You can do practically anything with this set. And if you like hemmed edges, you can do those using the provisional cast-on.
Oh lord, I didn't even notice that typo (and it's got to be a typo), I just read it as [sl1, k2tog, psso], which is pretty standard. Yeah, that would mess with your stitch count completely. I assumed a double decrease to match the double increase.
On the other hand, having 108st means your finished hat will look a lot more like the picture than if you'd actually worked the pattern as written. And if you continue the body of the hat in pattern, it will pretty much look exactly like it.
Seriously, though, this pattern is completely messed up.
Soozeq, by gauge I meant the difference in size between the loop when it's formed by wrapping around the yarn only, versus when it's formed by wrapping around a needle first. Not the gauge of the piece being worked.
I think we're talking at cross purposes and probably not using the same names for the techniques at this point, or perhaps working them slightly differently. Just sufficient to produce a visible difference. What I work as a long tail cast-on results in two rows, the cast-on row of very small loops, and a knit row. If I turn and knit the following row, it results in a line of purl bumps along my cast-on row on the front in stockinette. If I turn and purl, there is no such line.
When I work what I call the loop (or backwards loop, or "e") cast-on, turn and purl the row "tbl" (I'm using "tbl" to indicate that the stitches are worked as in Eastern or Combined orientation, with the leading edge of the stitch behind the needle tip), then it is indistinguishable from a long tail cast-on except that the foundation loops are looser. Turn and knit will result in an extra twist to each of the foundation stitches because the first stitch cannot be worked "tbl". However, if the first stitch is worked as usual, or slipped and the rest are knit "tbl", then again it is indistinguishable from long tail, except for the looseness of the foundation.
To make sure I'm not simply blowing gas, I swatched to double check and am confident that this is what I do. Works for me. The fuzziness of the terminology is a bit annoying, though.
I didn't notice any increases in the lace pattern (although the chart symbol used looks like an increase, which is misleading), but the decreases really should be worked over rows in pattern, as you said. Working it in stockinette will cause it to be much wider than the rest of the hat. Someone wasn't terribly careful in writing up this pattern.
It depends on how you work the loops of the loop cast-on. If you work them "through the back of the loop" which is actually the front (the leading edge is behind the needle tip, as in Eastern oriented stitches), it is indistinguishable except for the size of the restraining loops which you've mentioned. Difference in stitch size is more a result of gauge. It is inevitably going to be bigger when the yarn is wrapped around the needle instead of around the yarn. A lot of people end up with excess length in the CO row when they work the loop cast-on due to stressing it while working the first row and causing the loops to tighten up and slack between them to develop. Which is one reason the long tail cast-on is so much more regular looking.
"Favorite" is a problematic term, I think. I use the cast-on which I find most appropriate for the job. That said, the one I use most often when I'm not planning for anything specific is the long tail. These days, that's only about 1/3 of my knitting. Of all the cast-ons I use regularly, the one I use the absolute most is the invisible/provisional (not the one which is crocheted), but that's because I have specific plans for it.
The long tail cast-on is essentially the same thing as on loop cast-on followed by a worked row. The side which faces one as the cast on is being worked is the knit side, so if you turn it to work a flat piece, ideally it should be a purl row (for stockinette). The benefits of this method are that it's sturdy and stable while remaining elastic.
Personally? I'd frog it and work it up in a stitch which doesn't curl (seed, moss, mistake rib, etc). No need to worry about your steeking coming apart, and you wouldn't have to add an edging to it afterwards.
Also, extra practice in knitting leads to faster knitting overall, so it's not really wasted time.