Meanwhile we use a trick to get a full selection out of a partial one Direct Lasso makes — double Select > Inverse. I bound it to Ctrl+Shift+I, to match Photoshop’s hotkey, instead of spellcheck that uses it by default, and it works fine.
The request though is solid, thanks for remembering this. I wish I can add 10 votes to it.

Yes, you are quite right, there was such a tool, called just Lasso tool, and what we do have now as a Lasso used to be called Direct Select Lasso.
Take a look at the GIF attached.

I really wish for it to be back and extended, with an option to have only objects inside the line selected, along with the existing option to select all the objects inside and crossed by it.

Meanwhile, there is another request about freeform selection, a brush-like way: https://illustrator.uservoice.com/forums/333657/suggestions/38803612
And the plugin that allows that exists, if you are interested

Egor,
Back in AI10 there was DIrect lasso and Lasso.
Now only Direct lasso is in toolbox. Or is there a way to access "normal" lasso - when wanting to select full objects instead of just anchor points which is not the same?

Please see AI10 toolbox: https://www.oreilly.com/library/view/adobe-illustrator-10/0201756242/0201756242_ch01lev1sec3.html

Maciej, we have Lasso tool in Ai, still do.
Does is not get displayed for you in the toolbox?
Or you mean something else than a standard Lasso? You say 'Object Lasso' — is it a tool that selects whole objects instead of their anchors, like the normal Lasso does?

I have a trick for you. Select objects you want with Lasso, no matter which points. Select inverse. Select inverse again. There you have it—all object you partially selected with Lasso are now selected entirely.

I'm not sure under which tool this should exist but I need an option to help make whole selections. The (incorrectly named) Group (path!) selection tool enables you to select continuous paths but it's method for drawing is limited to rectangles. The Lasso tool on the other hand enables me to select in a freeform fashion but doesn't select the whole path, only anchors beneath the drawn out shape.

I'd like an option within the Lasso tool (when you double-click) which sets the selection behaviour like that of the Group Selection Tool, between path selection and anchor selection mode.

Oh and while you're at it, why not add an option to the Group Selection Tool so that it actually selects groups? Call it path selection and group selection mode.

Thanks!