After quite much research through the sparse documentation, and thanks to great questions and answers in SO, I came up with a simple class, that adds this functionality and and allows for configurations.

It looks like this:

You add this field in your grid like this:

Ext.define('Sandbox.view.OwnersGrid', {
    extend: 'Ext.grid.Panel',
    requires: ['Sandbox.view.SearchTrigger'],
    alias: 'widget.ownersGrid',
    store: 'Owners',
    columns: [{
        dataIndex: 'id',
        width: 50,
        text: 'ID'
    }, {
        dataIndex: 'name',
        text: 'Name',
    items:[{
        xtype: 'searchtrigger',
        autoSearch: true
    }]
},

The following configs are possible, and work like described in the doc for Ext.util.Filter:

• anyMatch
• caseSensitive
• exactMatch
• operator
• additionnaly you can use autoSearch. If true, the filter searches as you type, if false or not set, one has to click on the search icon to apply the filter.

ExtJs 5 / 6 Source:

Ext.define('Sandbox.view.SearchTrigger', {
    extend: 'Ext.form.field.Text',
    alias: 'widget.searchtrigger',
    triggers:{
        search: {
            cls: 'x-form-search-trigger',
            handler: function() {
                this.setFilter(this.up().dataIndex, this.getValue())
            }
        },
        clear: {
            cls: 'x-form-clear-trigger',
            handler: function() {
                this.setValue('')
                if(!this.autoSearch) this.setFilter(this.up().dataIndex, '')
            }
        }
    },
    setFilter: function(filterId, value){
        var store = this.up('grid').getStore();
        if(value){
            store.removeFilter(filterId, false)
            var filter = {id: filterId, property: filterId, value: value};
            if(this.anyMatch) filter.anyMatch = this.anyMatch
            if(this.caseSensitive) filter.caseSensitive = this.caseSensitive
            if(this.exactMatch) filter.exactMatch = this.exactMatch
            if(this.operator) filter.operator = this.operator
            console.log(this.anyMatch, filter)
            store.addFilter(filter)
        } else {
            store.filters.removeAtKey(filterId)
            store.reload()
        }
    },
    listeners: {
        render: function(){
            var me = this;
            me.ownerCt.on('resize', function(){
                me.setWidth(this.getEl().getWidth())
            })
        },
        change: function() {
            if(this.autoSearch) this.setFilter(this.up().dataIndex, this.getValue())
        }
    }
})

For ExtJs 6.2.0, the following bug and its workaround is relevant to this, else the column cannot be flexed.

ExtJs 4 Source:

Ext.define('Sandbox.view.SearchTrigger', {
    extend: 'Ext.form.field.Trigger',
    alias: 'widget.searchtrigger',
    triggerCls: 'x-form-clear-trigger',
    trigger2Cls: 'x-form-search-trigger',
    onTriggerClick: function() {
        this.setValue('')
        this.setFilter(this.up().dataIndex, '')
    },
    onTrigger2Click: function() {
        this.setFilter(this.up().dataIndex, this.getValue())
    },
    setFilter: function(filterId, value){
        var store = this.up('grid').getStore();
        if(value){
            store.removeFilter(filterId, false)
            var filter = {id: filterId, property: filterId, value: value};
            if(this.anyMatch) filter.anyMatch = this.anyMatch
            if(this.caseSensitive) filter.caseSensitive = this.caseSensitive
            if(this.exactMatch) filter.exactMatch = this.exactMatch
            if(this.operator) filter.operator = this.operator
            console.log(this.anyMatch, filter)
            store.addFilter(filter)
        } else {
            store.filters.removeAtKey(filterId)
            store.reload()
        }
    },
    listeners: {
        render: function(){
            var me = this;
            me.ownerCt.on('resize', function(){
                me.setWidth(this.getEl().getWidth())
            })
        },
        change: function() {
            if(this.autoSearch) this.setFilter(this.up().dataIndex, this.getValue())
        }
    }
})