Is there a definitive measure of completed lacto fermentation?

Well, if you really want a "definitive" way to measure when lactic fermentation is complete, you'd have to do a laboratory test directly for residual sugars. Most traditional foods that are lactic-acid fermented don't have enough sugar to reach a final acidity that will self-limit continued fermentation by killing all the bacteria. (Foods with too much sugar also often tend to spontaneously undergo yeast fermentation and lead to alcohol production instead if not monitored.) Given the wide variety of fermentation products that can result from initial ingredients and different fermentation conditions, it's really difficult to come up with some sort of "test strip" or something for residual sugar that is accurate. Home fermenters rarely bother with such things, but large-scale commercial fermentation sometimes tests residual sugar content for batches to ensure shelf-stability.

That said, a specific recipe with specific starting ingredients will often have a target pH range, depending on the quantity of sugar in the ingredients, the amount of salt (which in excess will inhibit further fermentation), desired flavor, etc. The composition of the ingredients will determine the mininum possible pH when fermentation is effectively complete.

The desired pH of lacto-fermented foods can vary anywhere from about 3.0 (even down to ~2.75 for some sweet relishes) to 4.5 (more common in grain-based ferments, or quick-fermented vegetables meant to be used up quickly). To answer the specific question for sauerkraut, I'd say most people would say sauerkraut fermentation is "complete" when a pH of ~3.5 or below is reached (though opinions may vary).

Part of the difficulty in answering the question is that home fermentation is often done with the intention of keeping the food "alive" even when it is consumed, which means fermentation will still be ongoing. In most traditional food cultures around the word, lactic acid fermentation is often deliberately incomplete, only used for a short period to transform an ingredient and then use it within a dish for consumption. This contrasts with much alcohol production, where continued fermentation in your final product can obviously create various difficulties for bottling, proper aging, etc.

And some people just prefer food at various earlier stages in the fermentation process. To take sauerkraut as an example, a pH of 4.0-4.5 will be achieved rather quickly (usually within a few days, depending on temperature, salt content, etc.). Some people prefer the taste of a fresher, crunchier product, and they may only ferment for a few days and then refrigerate to slow down fermentation significantly. Other people like to go much further and will ferment sauerkraut anywhere from a month to several months, again depending on temperature -- to achieve a final pH below 3.5 and even down to ~3.0. Over time, the final product will obviously get more sour, but it will also soften and various nutritional changes will take place. It's all a matter of preference when done at home.

Lastly, in reference to the "lagering" aspect of the question, I think the closest analog in lacto-fermentation is the fact that various processes (and various bacteria) will work better at different temperatures. A higher temperature fermentation, for example, may result in a greater acetic acid component in the final ingredients, enhancing the "sharp" vinegar notes. At lower temperatures other bacteria may become more prominent in the fermentation, and in some temperature ranges enzymes and natural (spoilage) reactions may help to change flavor and texture too. Anyhow, for these reasons, some people prefer to refrigerate fermented foods or hold them for a while at "cold cellar" temperatures to allow these "aging" mechanisms to work as bacteria slow down (analogous to lagering). Whether this is done after fermentation is effectively complete or at an earlier stage when the food is less sour again depends on individual preference.